Geometry. Chapter 7 Resource Masters

Transcription

1 Geometr hapter Resource Masters

2 onsumable Workbooks Man of the worksheets contained in the hapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook Skills Practice Workbook Practice Workbook Reading to Learn Mathematics Workbook NSWERS FOR WORKOOKS The answers for hapter of these workbooks can be found in the back of this hapter Resource Masters booklet. opright b The McGraw-Hill ompanies, Inc. ll rights reserved. Printed in the United States of merica. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced onl for classroom use; be provided to students, teachers, and families without charge; and be used solel in conjunction with Glencoe s Geometr. n other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill ompanies Orion Place olumbus, OH 30-0 ISN: Geometr hapter Resource Masters

3 ontents Vocabular uilder vii Proof uilder i Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson -3 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment hapter ssessment hapter Test, Form hapter Test, Form hapter Test, Form hapter Test, Form hapter Test, Form D hapter Test, Form hapter Open-Ended ssessment hapter Vocabular Test/Review hapter Quizzes & hapter Quizzes 3 & hapter Mid-hapter Test hapter umulative Review hapter Standardized Test Practice. Unit Test/Review (h. ) First Semester Test (h. ) Standardized Test Practice Student Recording Sheet NSWERS Glencoe/McGraw-Hill iii Glencoe Geometr

4 Teacher s Guide to Using the hapter Resource Masters The Fast File hapter Resource sstem allows ou to convenientl file the resources ou use most often. The hapter Resource Masters includes the core materials needed for hapter. These materials include worksheets, etensions, and assessment options. The answers for these pages appear at the back of this booklet. ll of the materials found in this booklet are included for viewing and printing in the Geometr TeacherWorks D-ROM. Vocabular uilder Pages vii viii include a student stud tool that presents up to twent of the ke vocabular terms from the chapter. Students are to record definitions and/or eamples for each term. You ma suggest that students highlight or star the terms with which the are not familiar. WHEN TO USE Give these pages to students before beginning Lesson -. Encourage them to add these pages to their Geometr Stud Notebook. Remind them to add definitions and eamples as the complete each lesson. Vocabular uilder Pages i include another student stud tool that presents up to fourteen of the ke theorems and postulates from the chapter. Students are to write each theorem or postulate in their own words, including illustrations if the choose to do so. You ma suggest that students highlight or star the theorems or postulates with which the are not familiar. WHEN TO USE Give these pages to students before beginning Lesson -. Encourage them to add these pages to their Geometr Stud Notebook. Remind them to update it as the complete each lesson. Stud Guide and Intervention Each lesson in Geometr addresses two objectives. There is one Stud Guide and Intervention master for each objective. WHEN TO USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN TO USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closel follow the structure of the Practice and ppl section of the Student Edition eercises. These eercises are of average difficult. WHEN TO USE These provide additional practice options or ma be used as homework for second da teaching of the lesson. Reading to Learn Mathematics One master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. dditional questions ask students to interpret the contet of and relationships among terms in the lesson. Finall, students are asked to summarize what the have learned using various representation techniques. WHEN TO USE This master can be used as a stud tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Glencoe/McGraw-Hill iv Glencoe Geometr

5 Enrichment There is one etension master for each lesson. These activities ma etend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics the are learning. These are not written eclusivel for honors students, but are accessible for use with all levels of students. WHEN TO USE These ma be used as etra credit, short-term projects, or as activities for das when class periods are shortened. ssessment Options The assessment masters in the hapter Resources Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. hapter ssessment HPTER TESTS Form contains multiple-choice questions and is intended for use with basic level students. Forms and contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms and D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with aes are provided for questions assessing graphing skills. Form 3 is an advanced level test with free-response questions. Grids without aes are provided for questions assessing graphing skills. ll of the above tests include a freeresponse onus question. The Open-Ended ssessment includes performance assessment tasks that are suitable for all students. scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. Vocabular Test, suitable for all students, includes a list of the vocabular words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate ssessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. Mid-hapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. ontinuing ssessment The umulative Review provides students an opportunit to reinforce and retain skills as the proceed through their stud of Geometr. It can also be used as a test. This master includes free-response questions. The Standardized Test Practice offers continuing review of geometr concepts in various formats, which ma appear on the standardized tests that the ma encounter. This practice includes multiplechoice, grid-in, and short-response questions. ubble-in and grid-in answer sections are provided on the master. nswers Page is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages This improves students familiarit with the answer formats the ma encounter in test taking. The answers for the lesson-b-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer kes are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe Geometr

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7 NME DTE PERIOD Reading to Learn Mathematics Vocabular uilder This is an alphabetical list of the ke vocabular terms ou will learn in hapter. s ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. dd these pages to our Geometr Stud Notebook to review vocabular at the end of the chapter. Vocabular Term ambiguous case Found on Page Definition/Description/Eample Vocabular uilder angle of depression angle of elevation cosine geometric mean Law of osines Law of Sines Pthagorean identit puh thag uh REE ahn (continued on the net page) Glencoe/McGraw-Hill vii Glencoe Geometr

8 NME DTE PERIOD Reading to Learn Mathematics Vocabular uilder (continued) Vocabular Term Pthagorean triple Found on Page Definition/Description/Eample reciprocal identit ri SIP ruh kuhl sine solve a triangle tangent trigonometric identit trig uh nuh MET rik trigonometric ratio trigonometr Glencoe/McGraw-Hill viii Glencoe Geometr

9 NME DTE PERIOD Learning to Read Mathematics Proof uilder This is a list of ke theorems and postulates ou will learn in hapter. s ou stud the chapter, write each theorem or postulate in our own words. Include illustrations as appropriate. Remember to include the page number where ou found the theorem or postulate. dd this page to our Geometr Stud Notebook so ou can review the theorems and postulates at the end of the chapter. Theorem or Postulate Theorem. Found on Page Description/Illustration/bbreviation Proof uilder Theorem. Theorem.3 Theorem. Pthagorean Theorem Theorem. onverse of the Pthagorean Theorem Theorem. Theorem. Glencoe/McGraw-Hill i Glencoe Geometr

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11 - NME DTE PERIOD Stud Guide and Intervention Geometric Mean Geometric Mean The geometric mean between two numbers is the square root of their product. For two positive numbers a and b, the geometric mean of a and b is the positive number in the proportion a. ross multipling gives b ab, so ab. Eample a. and 3 Find the geometric mean between each pair of numbers. Let represent the geometric mean or Definition of geometric mean ross multipl. Take the square root of each side. b. and Let represent the geometric mean Lesson - Eercises Find the geometric mean between each pair of numbers.. and. and 3. and 9. and. 3 and 3 3. and. 3 and. 0 and and 3 0. and. and. 3 and The geometric mean and one etreme are given. Find the other etreme. 3. is the geometric mean between a and b. Find b if a.. is the geometric mean between a and b. Find b if a 3. Determine whether each statement is alwas, sometimes, or never true.. The geometric mean of two positive numbers is greater than the average of the two numbers.. If the geometric mean of two positive numbers is less than, then both of the numbers are less than. Glencoe/McGraw-Hill 3 Glencoe Geometr

12 - NME DTE PERIOD Stud Guide and Intervention (continued) Geometric Mean ltitude of a Triangle In the diagram, D D. n altitude to the hpotenuse of a right triangle forms two right triangles. The two triangles are similar and each is similar to the original triangle. D Eample Eample Use right with D. Describe two geometric means. D a. D D so D D D. In, the altitude is the geometric mean between the two segments of the hpotenuse. b. D and D, so and. D D In, each leg is the geometric mean between the hpotenuse and the segment of the hpotenuse adjacent to that leg. Find,, and z. R PR PQ P Q PS z PR, PQ, PS Q ross multipl. 9 Divide each side b. Then PR SP 9 PR Q R Q R RS z PR, QR z, RS z z z z 00 ross multipl. z 0 Take the square root of each side. S P Eercises Find,, and z to the nearest tenth z z... z 3 z Glencoe/McGraw-Hill 3 Glencoe Geometr

13 - NME DTE PERIOD Skills Practice Geometric Mean Find the geometric mean between each pair of numbers. State eact answers and answers to the nearest tenth.. and. 9 and 3 3. and. and 0. and. 3 and Find the measure of each altitude. State eact answers and answers to the nearest tenth. Lesson -. D. P M L N 9. E H 0. S 9 G F R. U T Find and Glencoe/McGraw-Hill 33 Glencoe Geometr

14 NME DTE PERIOD - Practice Geometric Mean Find the geometric mean between each pair of numbers to the nearest tenth.. and. 3 and 3. and Find the measure of each altitude. State eact answers and answers to the nearest tenth.. U. J M T V L K Find,, and z... 3 z z z 0 z 0 0. IVIL ENGINEERING n airport, a factor, and a shopping center are at the vertices of a right triangle formed b three highwas. The airport and factor are.0 miles apart. Their distances from the shopping center are 3. miles and. miles, respectivel. service road will be constructed from the shopping center to the highwa that connects the airport and factor. What is the shortest possible length for the service road? Round to the nearest hundredth. Glencoe/McGraw-Hill 3 Glencoe Geometr

15 - NME DTE PERIOD Reading to Learn Mathematics Geometric Mean Pre-ctivit How can the geometric mean be used to view paintings? Read the introduction to Lesson - at the top of page 3 in our tetbook. What is a disadvantage of standing too close to a painting? What is a disadvantage of standing too far from a painting? Reading the Lesson. In the past, when ou have seen the word mean in mathematics, it referred to the average or arithmetic mean of the two numbers. a. omplete the following b writing an algebraic epression in each blank. If a and b are two positive numbers, then the geometric mean between a and b is Lesson - and their arithmetic mean is. b. Eplain in words, without using an mathematical smbols, the difference between the geometric mean and the algebraic mean.. Let r and s be two positive numbers. In which of the following equations is z equal to the geometric mean between r and s? s z. z r. r s r z z z. s: z z: r D. z s E. z r z s F. z s r z 3. Suppl the missing words or phrases to complete the statement of each theorem. a. The measure of the altitude drawn from the verte of the right angle of a right triangle to its hpotenuse is the between the measures of the two segments of the. b. If the altitude is drawn from the verte of the angle of a right triangle to its hpotenuse, then the measure of a is the of the of the triangle between the measure of the hpotenuse and the segment adjacent to that leg. c. If the altitude is drawn from the of the right angle of a right triangle to its Helping You Remember, then the two triangles formed are to the given triangle and to each other.. good wa to remember a new mathematical concept is to relate it to something ou alread know. How can the meaning of mean in a proportion help ou to remember how to find the geometric mean between two numbers? Glencoe/McGraw-Hill 3 Glencoe Geometr

16 - NME DTE PERIOD Enrichment Mathematics and Music Pthagoras, a Greek philosopher who lived during the sith centur.., believed that all nature, beaut, and harmon could be epressed b wholenumber relationships. Most people remember Pthagoras for his teachings about right triangles. (The sum of the squares of the legs equals the square of the hpotenuse.) ut Pthagoras also discovered relationships between the musical notes of a scale. These relationships can be epressed as ratios. D E F G When ou pla a stringed instrument, ou produce different notes b placing our finger on different places on a string. This is the result of changing the length of the vibrating part of the string. The string can be used to produce F b placing a finger 3 of the wa along the string. 3 of string Suppose a string has a length of inches. Write and solve proportions to determine what length of string would have to vibrate to produce the remaining notes of the scale.. D. E 3. F. G.... omplete to show the distance between finger positions on the -inch string for each note. For eample, () D in. D E F G 9. etween two consecutive musical notes, there is either a whole step or a half step. Using the distances ou found in Eercise, determine what two pairs of notes have a half step between them. Glencoe/McGraw-Hill 3 Glencoe Geometr

17 - NME DTE PERIOD Stud Guide and Intervention The Pthagorean Theorem and Its onverse The Pthagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hpotenuse. is a right triangle, so a b c. c b a Eample Prove the Pthagorean Theorem. With altitude D, each leg a and b is a geometric mean between hpotenuse c and the segment of the hpotenuse adjacent to that leg. c a a and c b b, so a c and b c. dd the two equations and substitute c to get a b c c c( ) c. Eample c D h a b a. Find a. a 3 a b c Pthagorean Theorem a 3 b, c 3 a 9 a a Simplif. Subtract. Take the square root of each side. b. Find c. 0 c 30 a b c Pthagorean Theorem 0 30 c a 0, b c Simplif. 300 c dd. 300 c Take the square root of each side. 3. c Use a calculator. Lesson - Eercises Find Glencoe/McGraw-Hill 3 Glencoe Geometr

18 - NME DTE PERIOD Stud Guide and Intervention (continued) The Pthagorean Theorem and Its onverse onverse of the Pthagorean Theorem If the sum of the squares of the measures of the two shorter sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. If the three whole numbers a, b, and c satisf the equation a b c, then the numbers a, b, and c form a Pthagorean triple. b If a b c, then is a right triangle. c a Eample Determine whether PQR is a right triangle. a b c Pthagorean Theorem 0 (0 3 ) 0 a 0, b 0 3, c Simplif dd. The sum of the squares of the two shorter sides equals the square of the longest side, so the triangle is a right triangle. R P 0 Q Eercises Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether the form a Pthagorean triple.. 30, 0, 0. 0, 30, 0 3.,, 30.,, 9. 3,,. 0,, 0.,, 3.,, 9. 9, 0, famil of Pthagorean triples consists of multiples of known triples. For each Pthagorean triple, find two triples in the same famil. 0. 3,,.,, 3.,, Glencoe/McGraw-Hill 3 Glencoe Geometr

19 - Find. NME DTE PERIOD Skills Practice The Pthagorean Theorem and Its onverse Determine whether STU is a right triangle for the given vertices. Eplain.. S(, ), T(, 3), U(3, ). S(3, 3), T(, ), U(, 0) Lesson - 9. S(, ), T(9, ), U(, 3) 0. S(0, 3), T(, ), U(, ). S( 3, ), T(, ), U(, ). S(, ), T(, ), U(, 3) Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether the form a Pthagorean triple. 3.,, 0., 30, 3.,, 0.,,.,,.,, Glencoe/McGraw-Hill 39 Glencoe Geometr

20 - Find. NME DTE PERIOD Practice The Pthagorean Theorem and Its onverse Determine whether GHI is a right triangle for the given vertices. Eplain.. G(, ), H(3, ), I(, ). G(, ), H(, ), I(, ) 9. G(, ), H(3, ), I(, ) 0. G(, ), H(, ), I(, 9) Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether the form a Pthagorean triple.. 9, 0,.,, 9 3., 3, 0. 9,, 3. 3,,., 3, 3. ONSTRUTION The bottom end of a ramp at a warehouse is 0 feet from the base of the main dock and is feet long. How high is the dock? ft ramp 0 ft? dock Glencoe/McGraw-Hill 30 Glencoe Geometr

21 NME DTE PERIOD - Reading to Learn Mathematics The Pthagorean Theorem and Its onverse Pre-ctivit How are right triangles used to build suspension bridges? Read the introduction to Lesson - at the top of page 30 in our tetbook. Do the two right triangles shown in the drawing appear to be similar? Eplain our reasoning. Reading the Lesson. Eplain in our own words the difference between how the Pthagorean Theorem is used and how the onverse of the Pthagorean Theorem is used.. Refer to the figure. For this figure, which statements are true?. m n p. n m p. m n p D. m p n E. p n m F. n p m m n p Lesson - G. n m p H. p m n 3. Is the following statement true or false? Pthagorean triple is an group of three numbers for which the sum of the squares of the smaller two numbers is equal to the square of the largest number. Eplain our reasoning.. If,, and z form a Pthagorean triple and k is a positive integer, which of the following groups of numbers are also Pthagorean triples?. 3,, z. 3, 3, 3z. 3, 3, 3z D. k, k, kz Helping You Remember. Man students who studied geometr long ago remember the Pthagorean Theorem as the equation a b c, but cannot tell ou what this equation means. formula is useless if ou don t know what it means and how to use it. How could ou help someone who has forgotten the Pthagorean Theorem remember the meaning of the equation a b c? Glencoe/McGraw-Hill 3 Glencoe Geometr

22 - NME DTE PERIOD Enrichment onverse of a Right Triangle Theorem You have learned that the measure of the altitude from the verte of the right angle of a right triangle to its hpotenuse is the geometric mean between the measures of the two segments of the hpotenuse. Is the converse of this theorem true? In order to find out, it will help to rewrite the original theorem in if-then form as follows. If Q is a right triangle with right angle at Q, then QP is the geometric mean between P and P, where P is between and and Q P is perpendicular to. Q P. Write the converse of the if-then form of the theorem.. Is the converse of the original theorem true? Refer to the figure at the right to eplain our answer. Q P You ma find it interesting to eamine the other theorems in hapter to see whether their converses are true or false. You will need to restate the theorems carefull in order to write their converses. Glencoe/McGraw-Hill 3 Glencoe Geometr

23 -3 NME DTE PERIOD Stud Guide and Intervention Special Right Triangles Properties of Triangles The sides of a right triangle have a special relationship. Eample Eample If the leg of a right triangle is units, show that the hpotenuse is units. Using the Pthagorean Theorem with a b, then c a b c Eercises Find. In a right triangle the hpotenuse is times the leg. If the hpotenuse is units, find the length of each leg. The hpotenuse is times the leg, so divide the length of the hpotenuse b. a 3 units Lesson Find the perimeter of a square with diagonal centimeters.. Find the diagonal of a square with perimeter 0 inches. 9. Find the diagonal of a square with perimeter meters. Glencoe/McGraw-Hill 33 Glencoe Geometr

24 -3 NME DTE PERIOD Stud Guide and Intervention (continued) Special Right Triangles Properties of Triangles The sides of a right triangle also have a special relationship. Eample In a right triangle, show that the hpotenuse is twice the shorter leg and the longer leg is 3 times the shorter leg. MNQ is a right triangle, and the length of the hpotenuse M N is two times the length of the shorter side N Q. Using the Pthagorean Theorem, a () 3 a 3 3 M a P 0 Q 0 N MNP is an equilateral triangle. MNQ is a right triangle. Eample In a right triangle, the hpotenuse is centimeters. Find the lengths of the other two sides of the triangle. If the hpotenuse of a right triangle is centimeters, then the length of the shorter leg is half of or. centimeters. The length of the longer leg is 3 times the length of the shorter leg, or (.)( 3 ) centimeters. Eercises Find and The perimeter of an equilateral triangle is 3 centimeters. Find the length of an altitude of the triangle to the nearest tenth of a centimeter.. n altitude of an equilateral triangle is.3 meters. Find the perimeter of the triangle to the nearest tenth of a meter. Glencoe/McGraw-Hill 3 Glencoe Geometr

25 -3 Find and. NME DTE PERIOD Skills Practice Special Right Triangles For Eercises 9, use the figure at the right.. If a, find b and c. c 0 a 30 b. If b, find a and c. 9. If c 9, find a and b. For Eercises 0 and, use the figure at the right. 0. The perimeter of the square is 30 inches. Find the length of. Lesson -3 D. Find the length of the diagonal D.. The perimeter of the equilateral triangle is 0 meters. Find the length of an altitude. E D 0 G F 3. GE is a triangle with right angle at E, and E is the longer leg. Find the coordinates of G in Quadrant I for E(, ) and (, ). Glencoe/McGraw-Hill 3 Glencoe Geometr

26 -3 Find and. NME DTE PERIOD Practice Special Right Triangles For Eercises 9, use the figure at the right.. If a 3, find b and c. c D 0 a. If 3 3, find a and D. 30 b 9. If a, find D, b, and. 0. The perimeter of an equilateral triangle is 39 centimeters. Find the length of an altitude of the triangle.. MIP is a triangle with right angle at I, and I P the longer leg. Find the coordinates of M in Quadrant I for I(3, 3) and P(, 3).. TJK is a triangle with right angle at J. Find the coordinates of T in Quadrant II for J(, 3) and K(3, 3). 3. OTNIL GRDENS One of the displas at a botanical garden is an herb garden planted in the shape of a square. The square measures ards on each side. Visitors can view the herbs from a diagonal pathwa through the garden. How long is the pathwa? d d d d Glencoe/McGraw-Hill 3 Glencoe Geometr

27 NME DTE PERIOD -3 Reading to Learn Mathematics Special Triangles Pre-ctivit How is triangle tiling used in wallpaper design? Read the introduction to Lesson -3 at the top of page 3 in our tetbook. How can ou most completel describe the larger triangle and the two smaller triangles in tile? How can ou most completel describe the larger triangle and the two smaller triangles in tile? (Include angle measures in describing all the triangles.) Reading the Lesson. Suppl the correct number or numbers to complete each statement. a. In a triangle, to find the length of the hpotenuse, multipl the length of a leg b. b. In a triangle, to find the length of the hpotenuse, multipl the length of the shorter leg b. c. In a triangle, the longer leg is opposite the angle with a measure of. d. In a triangle, to find the length of the longer leg, multipl the length of the shorter leg b. e. In an isosceles right triangle, each leg is opposite an angle with a measure of. f. In a triangle, to find the length of the shorter leg, divide the length of the longer leg b. g. In triangle, to find the length of the longer leg, divide the length of the hpotenuse b and multipl the result b. h. To find the length of a side of a square, divide the length of the diagonal b. Lesson -3. Indicate whether each statement is alwas, sometimes, or never true. a. The lengths of the three sides of an isosceles triangle satisf the Pthagorean Theorem. b. The lengths of the sides of a triangle form a Pthagorean triple. c. The lengths of all three sides of a triangle are positive integers. Helping You Remember 3. Some students find it easier to remember mathematical concepts in terms of specific numbers rather than variables. How can ou use specific numbers to help ou remember the relationship between the lengths of the three sides in a triangle? Glencoe/McGraw-Hill 3 Glencoe Geometr

28 -3 NME DTE PERIOD Enrichment onstructing Values of Square Roots The diagram at the right shows a right isosceles triangle with two legs of length inch. the Pthagorean Theorem, the length of the hpotenuse is inches. constructing an adjacent right triangle with legs of inches and inch, ou can create a segment of length 3. continuing this process as shown below, ou can construct a wheel of square roots. This wheel is called the Wheel of Theodorus after a Greek philosopher who lived about ontinue constructing the wheel until ou make a segment of length Glencoe/McGraw-Hill 3 Glencoe Geometr

29 - NME DTE PERIOD Stud Guide and Intervention Trigonometr Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan, respectivel. sin R leg opposite R leg adjacent to R cos R tan R hpotenuse hpotenuse r t s t r s S leg opposite R leg adjacent to R r T s t R Eample Find sin, cos, and tan. Epress each ratio as a decimal to the nearest thousandth. 3 sin o pposite leg hpotenuse cos a djacent leg hpotenuse opposite leg tan a djacent leg Eercises Find the indicated trigonometric ratio as a fraction and as a decimal. If necessar, round to the nearest ten-thousandth.. sin. tan 30 3 E D 0 F 3. cos. cos. sin D. tan E Lesson -. cos E. cos D Glencoe/McGraw-Hill 39 Glencoe Geometr

30 - NME DTE PERIOD Stud Guide and Intervention (continued) Trigonometr Use Trigonometric Ratios In a right triangle, if ou know the measures of two sides or if ou know the measures of one side and an acute angle, then ou can use trigonometric ratios to find the measures of the missing sides or angles of the triangle. Eample Find,, and z. Round each measure to the nearest whole number. z a. Find b. Find. tan tan tan 9 c. Find z. cos z cos z z cos z cos z 3 Eercises Find. Round to the nearest tenth Glencoe/McGraw-Hill 30 Glencoe Geometr

31 - NME DTE PERIOD Skills Practice Trigonometr Use RST to find sin R, cos R, tan R, sin S, cos S, and tan S. Epress each ratio as a fraction and as a decimal to the nearest hundredth. t S r. r, s 30, t 3. r 0, s, t R s T Use a calculator to find each value. Round to the nearest ten-thousandth. 3. sin. tan 3. cos. sin.. tan.3. cos.9 Use the figure to find each trigonometric ratio. Epress answers as a fraction and as a decimal rounded to the nearest ten-thousandth. 9. tan 0. sin. cos 9 0 Find the measure of each acute angle to the nearest tenth of a degree.. sin tan 0.. cos R 0.3. tan cos sin 0. Lesson - Find. Round to the nearest tenth L 33 9 S U Glencoe/McGraw-Hill 3 Glencoe Geometr

32 - NME DTE PERIOD Practice Trigonometr Use LMN to find sin L, cos L, tan L, sin M, cos M, and tan M. Epress each ratio as a fraction and as a decimal to the nearest hundredth. L., m 3, n 39., m 3, n N M Use a calculator to find each value. Round to the nearest ten-thousandth. 3. sin 9.. tan.. cos. Use the figure to find each trigonometric ratio. Epress answers as a fraction and as a decimal rounded to the nearest ten-thousandth.. cos. tan. sin 0 Find the measure of each acute angle to the nearest tenth of a degree. 9. sin tan 0.3. cos R 0.0 Find. Round to the nearest tenth GEOGRPHY Diego used a theodolite to map a region of land for his class in geomorpholog. To determine the elevation of a vertical rock formation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 3 meters and the angle was 3 degrees. What is the height of the formation to the nearest meter? 3 3 m Glencoe/McGraw-Hill 3 Glencoe Geometr

33 - NME DTE PERIOD Reading to Learn Mathematics Trigonometr Pre-ctivit How can surveors determine angle measures? Read the introduction to Lesson - at the top of page 3 in our tetbook. Wh is it important to determine the relative positions accuratel in navigation? (Give two possible reasons.) Reading the Lesson What does calibrated mean?. Refer to the figure. Write a ratio using the side lengths in the figure to represent each of the following trigonometric ratios. M N. sin N. cos N P. tan N D. tan M E. sin M F. cos M. ssume that ou enter each of the epressions in the list on the left into our calculator. Match each of these epressions with a description from the list on the right to tell what ou are finding when ou enter this epression. a. sin 0 b. cos 0 c. sin 0. d. tan 0. e. tan 0 f. cos 0. i. the degree measure of an acute angle whose cosine is 0. ii. the ratio of the length of the leg adjacent to the 0 angle to the length of hpotenuse in a triangle iii.the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length of the adjacent leg is 0. iv. the ratio of the length of the leg opposite the 0 angle to the length of the leg adjacent to it in a triangle v. the ratio of the length of the leg opposite the 0 angle to the length of hpotenuse in a triangle vi. the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length of the hpotenuse is 0. Lesson - Helping You Remember 3. How can the co in cosine help ou to remember the relationship between the sines and cosines of the two acute angles of a right triangle? Glencoe/McGraw-Hill 33 Glencoe Geometr

34 - NME DTE PERIOD Enrichment Sine and osine of ngles The following diagram can be used to obtain approimate values for the sine and cosine of angles from 0 to 90. The radius of the circle is. So, the sine and cosine values can be read directl from the vertical and horizontal aes Eample Find approimate values for sin 0 and cos 0. onsider the triangle formed b the segment marked 0, as illustrated b the shaded triangle at right. sin 0 a c 0. b or 0. cos 0 c 0. or c unit a sin b cos 0.. Use the diagram above to complete the chart of values sin 0. cos 0.. ompare the sine and cosine of two complementar angles (angles whose sum is 90 ). What do ou notice? Glencoe/McGraw-Hill 3 Glencoe Geometr

35 - NME DTE PERIOD Stud Guide and Intervention ngles of Elevation and Depression ngles of Elevation Man real-world problems that involve looking up to an object can be described in terms of an angle of elevation, which is the angle between an observer s line of sight and a horizontal line. line of sight angle of elevation Eample The angle of elevation from point to the top of a cliff is 3. If point is 000 feet from the base of the cliff, how high is the cliff? Let the height of the cliff. tan 3 tan o pposite 0 00 adjacent 000(tan 3 ) Multipl each side b Use a calculator ft The height of the cliff is about. feet. Eercises Solve each problem. Round measures of segments to the nearest whole number and angles to the nearest degree.. The angle of elevation from point to the top of a hill is 9. If point is 00 feet from the base of the hill, how high is the hill?? 9 00 ft. Find the angle of elevation of the sun when a.-meter-tall telephone pole casts a -meter-long shadow. sun? m. m 3. ladder leaning against a building makes an angle of with the ground. The foot of the ladder is feet from the building. How long is the ladder?? ft. person whose ees are feet above the ground is standing on the runwa of an airport 00 feet from the control tower. That person observes an air traffic controller at the window of the 3-foot tower. What is the angle of elevation? ft? 00 ft 3 ft Lesson - Glencoe/McGraw-Hill 3 Glencoe Geometr

36 - NME DTE PERIOD Stud Guide and Intervention (continued) ngles of Elevation and Depression ngles of Depression When an observer is looking down, the angle of depression is the angle between the observer s line of sight and a horizontal line. horizontal line of sight angle of depression Y Eample The angle of depression from the top of an 0-foot building to point on the ground is. How far is the foot of the building from point? Let the distance from point to the foot of the building. Since the horizontal line is parallel to the ground, the angle of depression D is congruent to. tan 0 tan o pposite adjacent (tan ) 0 Multipl each side b. 0 tan. Use a calculator. Divide each side b tan. Point is about 9 feet from the base of the building. Eercises horizontal D angle of depression 0 ft Solve each problem. Round measures of segments to the nearest whole number and angles to the nearest degree.. The angle of depression from the top of a sheer cliff to point on the ground is 3. If point is 0 feet from the base of the cliff, how tall is the cliff? 3? 0 ft. The angle of depression from a balloon on a -foot string to a person on the ground is 3. How high is the balloon? 3 ft? 3. ski run is 000 ards long with a vertical drop of 0 ards. Find the angle of depression from the top of the ski run to the bottom. 000 d? 0 d. From the top of a 0-foot-high tower, an air traffic controller observes an airplane on the runwa at an angle of depression of 9. How far from the base of the tower is the airplane?? 9 0 ft Glencoe/McGraw-Hill 3 Glencoe Geometr

37 - NME DTE PERIOD Skills Practice ngles of Elevation and Depression Name the angle of depression or angle of elevation in each figure... F L S T T S R W 3. D. Z R W P. MOUNTIN IKING On a mountain bike trip along the Gemini ridges Trail in Moab, Utah, Nabuko stopped on the canon floor to get a good view of the twin sandstone bridges. Nabuko is standing about 0 meters from the base of the canon cliff, and the natural arch bridges are about 00 meters up the canon wall. If her line of sight is five feet above the ground, what is the angle of elevation to the top of the bridges? Round to the nearest tenth degree.. SHDOWS Suppose the sun casts a shadow off a 3-foot building. If the angle of elevation to the sun is 0, how long is the shadow to the nearest tenth of a foot? 0? 3 ft. LLOONING From her position in a hot-air balloon, ngie can see her car parked in a field. If the angle of depression is and ngie is 3 meters above the ground, what is the straight-line distance from ngie to her car? Round to the nearest whole meter.. INDIRET MESUREMENT Kle is at the end of a pier 30 feet above the ocean. His ee level is 3 feet above the pier. He is using binoculars to watch a whale surface. If the angle of depression of the whale is 0, how far is the whale from Kle s binoculars? Round to the nearest tenth foot. whale 0 Kle s ees 3 ft pier 30 ft water level Lesson - Glencoe/McGraw-Hill 3 Glencoe Geometr

38 - NME DTE PERIOD Practice ngles of Elevation and Depression Name the angle of depression or angle of elevation in each figure.. T R. R P Z Y L M 3. WTER TOWERS student can see a water tower from the closest point of the soccer field at San Lobos High School. The edge of the soccer field is about 0 feet from the water tower and the water tower stands at a height of 3. feet. What is the angle of elevation if the ee level of the student viewing the tower from the edge of the soccer field is feet above the ground? Round to the nearest tenth degree.. ONSTRUTION roofer props a ladder against a wall so that the top of the ladder reaches a 30-foot roof that needs repair. If the angle of elevation from the bottom of the ladder to the roof is, how far is the ladder from the base of the wall? Round our answer to the nearest foot.. TOWN ORDINNES The town of elmont restricts the height of flagpoles to feet on an propert. Lindsa wants to determine whether her school is in compliance with the regulation. Her ee level is. feet from the ground and she stands 3 feet from the flagpole. If the angle of elevation is about, what is the height of the flagpole to the nearest tenth foot?. ft 3 ft. GEOGRPHY Stephan is standing on a mesa at the Painted Desert. The elevation of the mesa is about 30 meters and Stephan s ee level is. meters above ground. If Stephan can see a band of multicolored shale at the bottom and the angle of depression is 9, about how far is the band of shale from his ees? Round to the nearest meter.. INDIRET MESUREMENT Mr. Dominguez is standing on a 0-foot ocean bluff near his home. He can see his two dogs on the beach below. If his line of sight is feet above the ground and the angles of depression to his dogs are 3 and, how far apart are the dogs to the nearest foot? Mr. Dominguez ft 0 ft bluff 3 Glencoe/McGraw-Hill 3 Glencoe Geometr

39 - NME DTE PERIOD Reading to Learn Mathematics ngles of Elevation and Depression Pre-ctivit How do airline pilots use angles of elevation and depression? Read the introduction to Lesson - at the top of page 3 in our tetbook. What does the angle measure tell the pilot? Reading the Lesson. Refer to the figure. The two observers are looking at one another. Select the correct choice for each question. a. What is the line of sight? (i) line RS (ii) line ST (iii) line RT (iv) line TU b. What is the angle of elevation? (i) RST (ii) SRT (iii) RTS (iv) UTR c. What is the angle of depression? (i) RST (ii) SRT (iii) RTS (iv) UTR observer on ground R U observer at T top of building S d. How are the angle of elevation and the angle of depression related? (i) The are complementar. (ii) The are congruent. (iii) The are supplementar. (iv) The angle of elevation is larger than the angle of depression. e. Which postulate or theorem that ou learned in hapter 3 supports our answer for part c? (i) orresponding ngles Postulate (ii) lternate Eterior ngles Theorem (iii) onsecutive Interior ngles Theorem (iv) lternate Interior ngles Theorem. student sas that the angle of elevation from his ee to the top of a flagpole is 3. What is wrong with the student s statement? Helping You Remember 3. good wa to remember something is to eplain it to someone else. Suppose a classmate finds it difficult to distinguish between angles of elevation and angles of depression. What are some hints ou can give her to help her get it right ever time? Lesson - Glencoe/McGraw-Hill 39 Glencoe Geometr

40 - NME DTE PERIOD Enrichment Reading Mathematics The three most common trigonometric ratios are sine, cosine, and tangent. Three other ratios are the cosecant, secant, and cotangent. The chart below shows abbreviations and definitions for all si ratios. Refer to the triangle at the right. c a bbreviation Read as: Ratio leg opposite sin the sine of a c hpotenuse b leg adjacent to cos the cosine of b hpotenuse c leg opposite tan the tangent of a leg adjacent to b hpotenuse c csc the cosecant of leg opposite a hpotenuse c sec the secant of leg adjacent to b leg adjacent to cot the cotangent of b leg opposite a Use the abbreviations to rewrite each statement as an equation.. The secant of angle is equal to divided b the cosine of angle.. The cosecant of angle is equal to divided b the sine of angle. 3. The cotangent of angle is equal to divided b the tangent of angle.. The cosecant of angle multiplied b the sine of angle is equal to.. The secant of angle multiplied b the cosine of angle is equal to.. The cotangent of angle times the tangent of angle is equal to. Use the triangle at right. Write each ratio. R. sec R. csc R 9. cot R 0. sec S. csc S. cot S s t 3. If sin 0.9, find the value of csc. T r S. If tan.3, find the value of cot. Glencoe/McGraw-Hill 30 Glencoe Geometr

41 - NME DTE PERIOD Stud Guide and Intervention The Law of Sines The Law of Sines In an triangle, there is a special relationship between the angles of the triangle and the lengths of the sides opposite the angles. Law of Sines sin sin sin a b c Lesson - Eample Eample b 30 In, find b. sin sin Law of Sines c b sin sin m, c 30, m 30 b b sin 30 sin ross multipl. b 30 sin Divide each side b sin. sin b 0. Use a calculator. D E F sin D sin E d e si n D sin In DEF, find m D. Law of Sines d, m E, e sin D sin ross multipl. sin D s in D sin s in D. Divide each side b. Use the inverse sine. Use a calculator. Eercises Find each measure using the given measures of. Round angle measures to the nearest degree and side measures to the nearest tenth.. If c, m 0, and m 0, find a.. If b 0, c, and m, find m. 3. If a, c, and m, find m.. If a, m, and m, find b.. If b, m 9, and m 0, find a.. If a 30, c 0, and m 0, find m. Glencoe/McGraw-Hill 3 Glencoe Geometr

42 - NME DTE PERIOD Stud Guide and Intervention (continued) The Law of Sines Use the Law of Sines to Solve Problems You can use the Law of Sines to solve some problems that involve triangles. Law of Sines Let be an triangle with a, b, and c representing the measures of the sides opposite the angles with measures,, and, respectivel. Then sin a sin b sin c. Eample Isosceles has a base of centimeters and a verte angle of. Find the perimeter of the triangle. The verte angle is, so the sum of the measures of the base angles is and m m. sin sin b a Law of Sines c b a sin sin m, b, m a a sin sin a sin sin. ross multipl. Divide each side b sin. Use a calculator. The triangle is isosceles, so c.. The perimeter is.. or about centimeters. Eercises Draw a triangle to go with each eercise and mark it with the given information. Then solve the problem. Round angle measures to the nearest degree and side measures to the nearest tenth.. One side of a triangular garden is.0 feet. The angles on each end of this side measure and. Find the length of fence needed to enclose the garden.. Two radar stations and are 3 miles apart. The locate an airplane X at the same time. The three points form X, which measures, and X, which measures. How far is the airplane from each station? 3. civil engineer wants to determine the distances from points and to an inaccessible point in a river. measures and measures. If points and are.0 feet apart, find the distance from to each point.. ranger tower at point is kilometers north of a ranger tower at point. fire at point is observed from both towers. If measures 3 and measures, which ranger tower is closer to the fire? How much closer? Glencoe/McGraw-Hill 3 Glencoe Geometr

43 - NME DTE PERIOD Skills Practice The Law of Sines Find each measure using the given measures from. Round angle measures to the nearest tenth degree and side measures to the nearest tenth.. If m 3, m, and b, find a.. If m, m, and c, find b. Lesson - 3. If m, m, and a 3, find c.. If a, b, and m 3, find m.. If c 3, b 3, and m 3, find m.. If a, c 0, and m 3, find m.. If m, a, and m 0, find b. Solve each PQR described below. Round measures to the nearest tenth.. p, q 0, m P q, r, m R 0. p 9, q 3, m Q. If m P 9, p, r. If m Q 03, m P 3, p 3 3. If m P 9, m R, r 3. If m R 9, m Q, r. If m Q 3, m P, p 0. If q, m Q, m R. If r, p, m P Glencoe/McGraw-Hill 33 Glencoe Geometr

44 - NME DTE PERIOD Practice The Law of Sines Find each measure using the given measures from EFG. Round angle measures to the nearest tenth degree and side measures to the nearest tenth.. If m G, m E, and e, find g.. If e., m E, and m F, find f. 3. If g, f., and m G 3, find m F.. If e 9., m G 3, and m E, find g.. If f 9., g., and m G 3, find m F. Solve each STU described below. Round measures to the nearest tenth.. m T, s.3, t.. s 0, u, m S 3. m U 3, t.3, m T 9. m S, m U 9, s. 0. t., u., m T. m S 9, s.3, t. m T 9, m U, u t., m S, m T. INDIRET MESUREMENT To find the distance from the edge of the lake to the tree on the island in the lake, Hannah set up a triangular configuration as shown in the diagram. The distance from location to location is meters. The measures of the angles at and are and 3, respectivel. What is the distance from the edge of the lake at to the tree on the island at? Glencoe/McGraw-Hill 3 Glencoe Geometr

45 - NME DTE PERIOD Reading to Learn Mathematics The Law of Sines Pre-ctivit How are triangles used in radio astronom? Read the introduction to Lesson - at the top of page 3 in our tetbook. Wh might several antennas be better than one single antenna when studing distant objects? Lesson - Reading the Lesson. Refer to the figure. ccording to the Law of Sines, which of the following are correct statements? m n p.. sin m si n n sin p sin M sin N sin P M N P. co s M cos N co s P m n p sin M D. sin N sin P m n p M n P p m N E. (sin M) (sin N) (sin P) F. sin P sin M sin N p m n. State whether each of the following statements is true or false. If the statement is false, eplain wh. a. The Law of Sines applies to all triangles. b. The Pthagorean Theorem applies to all triangles. c. If ou are given the length of one side of a triangle and the measures of an two angles, ou can use the Law of Sines to find the lengths of the other two sides. d. If ou know the measures of two angles of a triangle, ou should use the Law of Sines to find the measure of the third angle. e. friend tells ou that in triangle RST, m R 3, r centimeters, and s 3 centimeters. an ou use the Law of Sines to solve the triangle? Eplain. Helping You Remember 3. Man students remember mathematical equations and formulas better if the can state them in words. State the Law of Sines in our own words without using variables or mathematical smbols. Glencoe/McGraw-Hill 3 Glencoe Geometr

46 - NME DTE PERIOD Enrichment Identities n identit is an equation that is true for all values of the variable for which both sides are defined. One wa to verif an identit is to use a right triangle and the definitions for trigonometric functions. c a Eample is an identit. Verif that (sin ) (cos ) b (sin ) (cos ) a c b c b a c c c To check whether an equation ma be an identit, ou can test several values. However, since ou cannot test all values, ou cannot be certain that the equation is an identit. Eample Test sin sin cos to see if it could be an identit. Tr 0. Use a calculator to evaluate each epression. sin sin 0 sin cos (sin 0)(cos 0) 0.3 (0.3)(0.90) 0.3 Since the left and right sides seem equal, the equation ma be an identit. Use triangle shown above. Verif that each equation is an identit.. c os sin tan. t an sin co s 3. tan cos sin. (cos ) (sin ) Tr several values for to test whether each equation could be an identit.. cos (cos ) (sin ). cos (90 ) sin Glencoe/McGraw-Hill 3 Glencoe Geometr

47 - NME DTE PERIOD Stud Guide and Intervention The Law of osines The Law of osines nother relationship between the sides and angles of an triangle is called the Law of osines.you can use the Law of osines if ou know three sides of a triangle or if ou know two sides and the included angle of a triangle. Law of osines Let be an triangle with a, b, and c representing the measures of the sides opposite the angles with measures,, and, respectivel. Then the following equations are true. a b c bc cos b a c ac cos c a b ab cos Eample In, find c. c a b ab cos Law of osines c 0 ()(0)cos a, b 0, m c 0 (0)co () s c 9. Use a calculator. Take the square root of each side. 0 c Lesson - Eample In, find m. a b c bc cos Law of osines ()() cos a, b, c 9 0 cos Multipl. 0 0 cos Subtract 9 from each side. cos Divide each side b 0. cos Use the inverse cosine. 0 Use a calculator. Eercises Find each measure using the given measures from. Round angle measures to the nearest degree and side measures to the nearest tenth.. If b, c, and m, find a.. If a, b 0, and c, find m. 3. If a, b, and c, find m.. If a 0, c, and m, find b.. If b, c, and m 9, find a.. If a, b 9, and c, find m. Glencoe/McGraw-Hill 3 Glencoe Geometr

48 - NME DTE PERIOD Stud Guide and Intervention (continued) The Law of osines Use the Law of osines to Solve Problems You can use the Law of osines to solve some problems involving triangles. Law of osines Let be an triangle with a, b, and c representing the measures of the sides opposite the angles with measures,, and, respectivel. Then the following equations are true. a b c bc cos b a c ac cos c a b ab cos Eample Ms. Jones wants to purchase a piece of land with the shape shown. Find the perimeter of the propert. Use the Law of osines to find the value of a. a b c bc cos Law of osines a (300)(00) cos b 300, c 00, m a 30,0 00 0,0 00 cos 3. Take the square root of each side. Use a calculator. Use the Law of osines again to find the value of c. c a b ab cos Law of osines c (3.)(300) cos 0 a 3., b 300, m 0 c,.09,0 0 cos.9 Take the square root of each side. Use a calculator. The perimeter of the land is or about 3 feet. Eercises 300 ft 0 a 300 ft 00 ft c Draw a figure or diagram to go with each eercise and mark it with the given information. Then solve the problem. Round angle measures to the nearest degree and side measures to the nearest tenth.. triangular garden has dimensions feet, feet, and feet. Find the angles at each corner of the garden.. parallelogram has a angle and sides and. Find the lengths of the diagonals. 3. n airplane is sighted from two locations, and its position forms an acute triangle with them. The distance to the airplane is 0 miles from one location with an angle of elevation.0, and 0 miles from the other location with an angle of elevation of.. How far apart are the two locations?. ranger tower at point is directl north of a ranger tower at point. fire at point is observed from both towers. The distance from the fire to tower is 0 miles, and the distance from the fire to tower is 0 miles. If m, find the distance between the towers. Glencoe/McGraw-Hill 3 Glencoe Geometr

49 - NME DTE PERIOD Skills Practice The Law of osines In RST, given the following measures, find the measure of the missing side.. r, s, m T 39. r, t, m S 3. r 9, t, m S 03. s, t 0, m R In HIJ, given the lengths of the sides, find the measure of the stated angle to the nearest tenth.. h, i, j ; m H Lesson -. h, i, j ; m I. h 3, i, j 9; m J. h 3, i, j 30; m H Determine whether the Law of Sines or the Law of osines should be used first to solve each triangle. Then solve each triangle. Round angle measures to the nearest degree and side measures to the nearest tenth c 9 33 M L N. a 0, b, c 9. a, b 0, m Solve each RST described below. Round measures to the nearest tenth. 3. r, s 3, t 3. r 30, s, m T. r, s, m R. r, s, t 30 Glencoe/McGraw-Hill 39 Glencoe Geometr

50 - NME DTE PERIOD Practice The Law of osines In JKL, given the following measures, find the measure of the missing side.. j.3, k 0, m L. j 9.,., m K 3 3. j, k, m L 3. k.,., m J In MNQ, given the lengths of the sides, find the measure of the stated angle to the nearest tenth.. m, n 3, q ; m Q. m, n, q 3; m M. m.9, n, q 0.; m N. m 3, n 30., q ; m Q Determine whether the Law of Sines or the Law of osines should be used first to solve. Then sole each triangle. Round angle measures to the nearest degree and side measure to the nearest tenth. 9. a 3, b, c 9 0. a, b 9, m 3. a, b, m 9. a., b, m Solve each FGH described below. Round measures to the nearest tenth. 3. m F, f., g. f 0, g 3, m H. f., g, h. f 3, h 30, m G. REL ESTTE The Esposito famil purchased a triangular plot of land on which the plan to build a barn and corral. The lengths of the sides of the plot are 30 feet, feet, and 30 feet. What are the measures of the angles formed on each side of the propert? Glencoe/McGraw-Hill 390 Glencoe Geometr

51 - NME DTE PERIOD Reading to Learn Mathematics The Law of osines Pre-ctivit How are triangles used in building design? Read the introduction to Lesson - at the top of page 3 in our tetbook. What could be a disadvantage of a triangular room? Reading the Lesson. Refer to the figure. ccording to the Law of osines, which statements are correct for DEF?. d e f ef cos D. e d f df cos E. d e f ef cos D D. f d e ef cos F E. f d e de cos F F. d e f G. sin D sin E sin F d e f H. d e f f e D cos D e F f d E Lesson -. Each of the following describes three given parts of a triangle. In each case, indicate whether ou would use the Law of Sines or the Law of osines first in solving a triangle with those given parts. (In some cases, onl one of the two laws would be used in solving the triangle.) a. SSS b. S c. S d. SS e. SS 3. Indicate whether each statement is true or false. If the statement is false, eplain wh. a. The Law of osines applies to right triangles. b. The Pthagorean Theorem applies to acute triangles. c. The Law of osines is used to find the third side of a triangle when ou are given the measures of two sides and the nonincluded angle. d. The Law of osines can be used to solve a triangle in which the measures of the three sides are centimeters, centimeters, and centimeters. Helping You Remember. good wa to remember a new mathematical formula is to relate it to one ou alread know. The Law of osines looks somewhat like the Pthagorean Theorem. oth formulas must be true for a right triangle. How can that be? Glencoe/McGraw-Hill 39 Glencoe Geometr

52 - NME DTE PERIOD Enrichment Spherical Triangles Spherical trigonometr is an etension of plane trigonometr. Figures are drawn on the surface of a sphere. rcs of great circles correspond to line segments in the plane. The arcs of three great circles intersecting on a sphere form a spherical triangle. ngles have the same measure as the tangent lines drawn to each great circle at the verte. Since the sides are arcs, the too can be measured in degrees. b c a The sum of the sides of a spherical triangle is less than 30. The sum of the angles is greater than 0 and less than 0. The Law of Sines for spherical triangles is as follows. sin a sin b sin c s in s in s in There is also a Law of osines for spherical triangles. cos a cos b cos c sin b sin c cos cos b cos a cos c sin a sin c cos cos c cos a cos b sin a sin b cos Eample Solve the spherical triangle given a, b 0, and c. Use the Law of osines ( 0.)(0.) (0.99)(0.) cos cos (0.3090)(0.) (0.9)(0.) cos cos (0.3090)( 0.) (0.9)(0.99) cos cos 0. heck b using the Law of Sines. s in s in 0 s in. sin 9 sin 9 sin Solve each spherical triangle.. a, b 3, c 9. a 0, b 33, c 9 3. a, b 0, 9. b 9, c, Glencoe/McGraw-Hill 39 Glencoe Geometr

53 NME DTE PERIOD hapter Test, Form SORE Write the letter for the correct answer in the blank at the right of each question.. Find the geometric mean between 0 and D. 0.. Find in D.. 3. Find in PQR D. 0 R P Q 3.. Find in STU... U. 3 D. S. Which set of measures could represent the sides of a right triangle?., 3,.,,., 0, D. 9,, T.. ssessments. Find in DEF D. F D E.. Find in XYZ D. 30 Z X Y.. The length of the sides of a square is 0 meters. Find the length of the diagonal of the square.. 0 m. 0 m. 0 3 m D. 0 m. 9. Find in HJK D. K 0 H 30 J Find in D Glencoe/McGraw-Hill 393 Glencoe Geometr

54 NME DTE PERIOD hapter Test, Form (continued). In QRS, R is a right angle. Which is the ratio for the tangent of S? measure of leg adjacent to S. measure of hpotenuse. measure of leg opposite S. measure of hpotenuse D. measure of hpotenuse measure of leg opposite S measure of leg opposite S measure of leg adjacent to S.. Find cos in.... D.. 3. Find to the nearest tenth D.. 0. Find the angle of elevation of the sun when a pole feet tall casts a shadow feet long D Which is the angle of depression in the figure at the right?. OT. O. TO D. TO O T.. Find in XYZ to the nearest tenth if m Y 3, m X 9, and D... To find the distance between two points and on opposite sides of a river, a surveor measures the distance from to as 00 feet, m, and m 3. Find the distance from to. Round our answer to the nearest tenth... ft. 0. ft. 0. ft D. 3. ft. In, a, b, and m 0. Find m to the nearest tenth D.. 9. Find the third side of a triangular garden if two sides are feet and feet and the included angle has a measure 0... ft. 9. ft.. ft D..3 ft 0. In DEF, d 0, e, and f 30. Find m F to the nearest tenth D onus In, a 0, b, and c 0. Find m to the nearest tenth. : Glencoe/McGraw-Hill 39 Glencoe Geometr

55 NME DTE PERIOD hapter Test, Form SORE Write the letter for the correct answer in the blank at the right of each question.. Find the geometric mean between and D... In PQR, RS and QS. Find PS D. R S P Q. 3. Find..... D Find D.. Find the length of the hpotenuse of a right triangle whose legs measure and D. 3.. ssessments. Find D... Which of the following could represent sides of a right triangle?. 9, 0,., 30, 3.,, D., 3,.. Find c D. c. 9. Find the perimeter of a square to the nearest tenth if the length of its diagonal is inches... in in.. in. D..9 in Find D Glencoe/McGraw-Hill 39 Glencoe Geometr

56 NME DTE PERIOD hapter Test, Form (continued). Find to the nearest tenth D In right triangle, a, b 9, and c. Find tan D Find to the nearest tenth of a degree D If a 0-foot ladder makes a angle with the ground, how man feet up a wall will it reach? Round our answer to the nearest tenth... ft. 0 ft.. ft D..9 ft. ship s sonar finds that the angle of depression to a wreck on the bottom of the ocean is.. If a point on the ocean floor is 0 meters directl below the ship, how man meters, to the nearest tenth, is it from that point on the ocean floor to the wreck?.. m. 0. m.. m D. 3.3 m. To the nearest tenth of a degree, find the angle of elevation of the sun if a building 00 feet tall casts a shadow 0 feet long D When the sun s angle of elevation is 3, a tree tilted at an angle of from the vertical casts a 0-foot shadow on the ground. Find the length of the tree to the nearest tenth...3 ft. 9. ft.. ft D. 9. ft 3 0-foot shadow.. In DE, m, m D, and e.. Find c to the nearest tenth D.. 9. In PQR, p, r, and m Q 0. Find q to the nearest tenth D Pete is building a kite using the dimensions given in the figure at the right. Find the measure of the angle the -foot edge makes with the 3-foot edge D.. ft ft ft 3 ft 3 ft 0. onus From a window 0 feet above the ground, the angle of elevation to the top of another building is 3. The distance between the buildings is feet. Find the height of the building to the nearest tenth of a foot. : Glencoe/McGraw-Hill 39 Glencoe Geometr

57 NME DTE PERIOD hapter Test, Form SORE Write the letter for the correct answer in the blank at the right of each question.. Find the geometric mean between 9 and D.. In PQR, RS and QS. Find PS D Find..... D. 33. Find.... D. 9. Find the length of the hpotenuse of a right triangle whose legs measure and D.. Find D. R P 3 S Q ssessments. Which of the following could represent sides of a right triangle?. 3,,. 3,,.,, D.,,.. Find c D. 9 9 c. 9. Find the perimeter of a square to the nearest tenth if the length of its diagonal is millimeters...3 mm..3 mm. 90. mm D..0 mm 0. Find D Glencoe/McGraw-Hill 39 Glencoe Geometr

58 NME DTE PERIOD hapter Test, Form (continued). Find D In right triangle, a, b, and c 0. Find tan.... D.. 3. Find to the nearest tenth of a degree D If a -foot ladder makes a angle with the ground, how man feet up a wall will it reach? Round our answer to the nearest tenth.. 3. ft. 0. ft. 0. ft D.. ft. ship s sonar finds that the angle of depression to a wreck on the bottom of the ocean is 3.. If a point on the ocean floor is meters directl below the ship, how man meters, to the nearest tenth, is it from that point on the ocean floor to the wreck?. 3. m. 39. m..0 m D.. m. To the nearest tenth of a degree, find the angle of elevation of the sun if a building feet tall casts a shadow 9 feet long D When the sun s angle of elevation is, a tree tilted at an angle of from the vertical casts a -foot shadow on the ground. Find the length of the tree, to the nearest tenth.. 0. ft.. ft.. ft D.. ft -foot shadow.. In, m, m 0, and c 9.. Find a to the nearest tenth D.. 9. In LMN, l, m, and m N 0. Find n to the nearest tenth D.. 0. Josephine is planning a triangular garden. If the lengths of the sides are 0 feet, 0 feet, and 00 feet, what is the measure of the largest angle? D onus From a window feet above the ground, the angle of elevation to the top of another building is 3. The distance between the buildings is 3 feet. Find the height of the building to the nearest tenth of a foot. : Glencoe/McGraw-Hill 39 Glencoe Geometr

59 NME DTE PERIOD hapter Test, Form SORE. Find the geometric mean between and. For Questions, find Determine whether is a right triangle. Eplain our answer... Find. O. ssessments. In parallelogram D, D and m D 0. Find F. F D. 9. Find and Find to the nearest tenth n -frame house is 0 feet high and 30 feet wide. Find the measure of the angle, to the nearest tenth of a degree, that the roof makes with the floor. 0 ft 30 ft.. 30-foot tree casts a -foot shadow. Find the angle of elevation of the sun to the nearest tenth of a degree.. Glencoe/McGraw-Hill 399 Glencoe Geometr

60 NME DTE PERIOD hapter Test, Form (continued) 3. boat is 000 meters from a cliff. If the angle of depression from the top of the cliff to the boat is, how tall is the cliff? Round our answer to the nearest tenth. 000 m 3.. plane fling at an altitude of 0,000 feet begins descending when the end of the runwa is below a point 0,000 feet awa. Find the angle of descent (depression) to the nearest tenth of a degree... Find to the nearest tenth.. 3. Find to the nearest tenth of a degree. 3.. tree grew at a 3 slant from the vertical. t a point 0 feet from the tree, the angle of elevation to the top of the tree is. Find the length of the tree to the nearest tenth of a foot. 0 ft 93.. Find to the nearest tenth of a degree.. 9. In XYZ, m X,, and z 9. Find to the nearest tenth To approimate the length of a pond, a surveor walks 00 meters from point L to point K, then turns and walks 0 meters from point K to point E. If m LKE 0, find the length LE of the pond to the nearest tenth of a meter. L K 00 m 0 m 0 E 0. onus Find. : Glencoe/McGraw-Hill 00 Glencoe Geometr

61 NME DTE PERIOD hapter Test, Form D SORE. Find the geometric mean between 3 and.. For Questions, find Determine whether is a right triangle. Eplain our answer... Find. O 30. ssessments. In parallelogram D, D and m D 0. Find F. F D. 9. Find and Find to the nearest tenth n -frame house is feet high and 3 feet wide. Find the measure of the angle, to the nearest tenth of a degree, that the roof makes with the floor. ft 3 ft.. 3-foot tree casts a -foot shadow. Find the angle of elevation of the sun to the nearest tenth of a degree.. Glencoe/McGraw-Hill 0 Glencoe Geometr

62 NME DTE PERIOD hapter Test, Form D (continued) 3. boat is 000 meters from a cliff. If the angle of depression from the top of the cliff to the boat is 0, how tall is the cliff? Round our answer to the nearest tenth. 000 m 3.. plane fling at an altitude of 0,000 feet begins descending when the end of the runwa is below a point 0,000 feet awa. Find the angle of descent (depression) to the nearest tenth of a degree... Find to the nearest tenth. 3.. Find to the nearest tenth of a degree. 3.. tree grew at a 3 slant from the vertical. t a point 0 feet from the tree, the angle of elevation to the top of the tree is. Find the length of the tree to the nearest tenth of a foot. 0 ft 93.. Find to the nearest tenth of a degree In XYZ, m X,, and z. Find to the nearest tenth To approimate the length of a pond, a surveor walks 0 meters from point L to point K, then turns and walks 0 meters from point K to point E. If m LKE, find the length LE of the pond to the nearest tenth of a meter. L K 0 m 0 m E 0. onus Find. : Glencoe/McGraw-Hill 0 Glencoe Geometr

63 NME DTE PERIOD hapter Test, Form 3 SORE. Find the geometric mean between 9 and Find in PQR. P S. Q R 3. Find in XYZ. Y 3. X W Z. If the length of one leg of a right triangle is three times the length of the other and the hpotenuse is 0, find the length of the shorter leg... Find the length of the altitude to the hpotenuse of a right triangle with legs of length 3 and.. Find ssessments. Richmond is 00 kilometers due east of Teratown and Hamilton is 0 kilometers directl north of Teratown. Find the shortest distance in kilometers between Hamilton and Richmond.. Is,, 3 a Pthagorean triple? Show wh or wh not Find the perimeter of this square Find the perimeter of rectangle D. D Find and is a triangle with right angle and with as the longer leg. Find the coordinates of if (, ) and (, ).. Glencoe/McGraw-Hill 03 Glencoe Geometr

64 NME DTE PERIOD hapter Test, Form 3 (continued) 3. If D, find and the length of D D. The angle of elevation from a point on the street to the top of a building is 9. The angle of elevation from another point on the street, 0 feet farther awa from the building, to the top of the building is. To the nearest foot, how tall is the building?.. The angle of depression from the top of a flagpole on top of a lighthouse to a boat on the ocean is 3, while the angle of depression from the bottom of the flagpole to the boat is 3.. If the boat is mile awa from shore and the lighthouse is right on the edge of the shore, how tall is the flagpole? Round our answer to the nearest foot... In JKL, m J., m K 9, and k. Find to the nearest tenth... Solve PQR for r, p, and m Q 9. Round answers to the nearest tenth... Don hit a golf ball 0 ards toward the hole. However, due to the wind, his drive is off course. If the angle between the hole and where the ball lands is 9, how far is it from where Don hit the ball to the hole? Round to the nearest tenth of a ard.. 9. In HJK, m H 3, k, and h. Find m K. Round our answer(s) to the nearest tenth of a degree The distance from lban to ethan is miles and from ethan to elina 0 miles. If the roads from ethan to lban and from ethan to elina make an angle, what is the distance from lban to elina? Round to the nearest tenth. 0. onus 0-foot vertical pole that stands on a hillside makes an angle of 0 with the horizontal. Two gu wires etend from the top of the pole to points on the hill 0 feet uphill and downhill from its base. Find the length of each gu wire to the nearest tenth. : Glencoe/McGraw-Hill 0 Glencoe Geometr

65 NME DTE PERIOD hapter Open-Ended ssessment SORE Demonstrate our knowledge b giving a clear, concise solution to each problem. e sure to include all relevant drawings and justif our answers. You ma show our solution in more than one wa or investigate beond the requirements of the problem.. If the geometric mean between 0 and is, what is? Show how ou obtained our answer.. R P 0 S Q a. Ma used the following equations to find in PQR. Is Ma correct? Wh or wh not? ssessments b. For PRQ to be a right angle, what would the measure of P S have to be? c. Is PRS a triangle? How do ou know? 3. To solve for in a triangle, when would ou use sin and when would ou use sin? Give an eample for each tpe of situation.. Draw a diagram showing where the angles of elevation and depression are. How are the measures of these angles related?. Draw an obtuse triangle and label the vertices and the measures of two angles and the length of one side. Eplain how to solve the triangle.. Irina is solving. She plans to first use the Law of Sines to find two of the angles. Is Irina s plan a good one? Wh or wh not? Glencoe/McGraw-Hill 0 Glencoe Geometr

66 NME DTE PERIOD hapter Vocabular Test/Review SORE ambiguous case angle of depression angle of elevation cosecant cosine geometric mean Law of osines Law of Sines Pthagorean identit Pthagorean triple reciprocal identities secant sine solving a triangle tangent trigonometric identit trigonometric ratio trigonometr hoose from the terms above to complete each sentence.. The square root of the product of two numbers is the of the numbers.?.. group of three whole numbers that satisf the equation a b c, where c is the greatest number, is called a(n)?. 3. ratio of the lengths of two sides of a right triangle is called a(n)?.. n angle between the line of sight and the horizontal when an observer looks upward is called a(n)?.. n angle between the line of sight and the horizontal when an observer looks downward is called a(n)?.. Three commonl used trigonometric ratios are the?,?, and?.. For, the? sas sin sin sin. a b c. For, the? sas a b c bc cos. 9. The reciprocal of the sine is called the?. 0. The reciprocal of the cosine is called the?. Define each term.. solving a triangle Pthagorean Theorem. Glencoe/McGraw-Hill 0 Glencoe Geometr

67 NME DTE PERIOD hapter Quiz (Lessons and ) SORE. Find the geometric mean between and.. For Questions and 3, find and Find... Do 9,, and 3 form a Pthagorean triple? Wh or wh not?. hapter Quiz (Lessons 3 and ) For Questions and, find... For Questions 3 and, find to the nearest tenth. 3.. NME DTE PERIOD SORE ssessments. rectangle has a diagonal 0 inches long that forms angles of 0 and 30 with the sides. Find the perimeter of the rectangle.. Find sin. Round to the nearest ten-thousandth.. If cos 0.9, find to the nearest tenth of a degree.. The distance along a hill is feet. If the land slopes uphill at an angle of, find the vertical distance from the top to the bottom of the hill. 9. surveor is standing on horizontal ground level with the base of a skscraper. The angle formed b the line segment from his position to the top of the skscraper is 3. The height of the building is 00 feet. Find the distance from the building to the surveor to the nearest foot Glencoe/McGraw-Hill 0 Glencoe Geometr

68 NME DTE PERIOD hapter Quiz (Lessons and ) SORE. Name the angle of elevation in the figure. S R. P Q. Find to the nearest tenth Solve. Round our answers to the nearest tenth triangular lot has 00 feet of frontage along a river. The other two sides make angles of and with the riverfront side. Find the length of the shortest side to the nearest foot.. STNDRDIZED TEST PRTIE squirrel 3 feet up in a tree sees a dog 9 feet from the base of the tree. Find the measure of the angle of depression to the nearest tenth of a degree D.... NME DTE PERIOD hapter Quiz (Lesson ) SORE For Questions and, find to the nearest tenth Solve RST. Round our answers to the nearest degree. R T S 3.. hiker is miles from a tower and miles from the lodge. She estimates that the angle between her path to the tower and her path to the lodge is. Find the distance from the tower to the lodge to the nearest tenth of a mile.. STNDRDIZED TEST PRTIE For, find a to the nearest tenth if m 9, b, and c D Glencoe/McGraw-Hill 0 Glencoe Geometr

69 NME DTE PERIOD hapter Mid-hapter Test (Lessons through ) SORE Part I Write the letter for the correct answer in the blank at the right of each question.. Find the geometric mean between and D.. Find D Find sin D Find to the nearest tenth D ssessments. Find to the nearest tenth of a degree D Part II For Questions, find and Do, 90, 0 form a Pthagorean triple? Wh or wh not? 0. Gu wires 0 feet long support a -foot tall telephone pole. To the nearest tenth of a degree, what angle will the wires make with the ground? Glencoe/McGraw-Hill 09 Glencoe Geometr

70 NME DTE PERIOD hapter umulative Review (hapters ) SORE. Name the verte and sides, then classif IFT. (Lesson -) I 9 F T. For Questions and 3, complete the following proof. (Lesson -) Given: J K L M H J K L Prove: H K K M Proof: Statements Reasons. J K L M, H J K L. Given. JK LM, HJ KL. (Question ) 3. (Question 3) 3. Segment ddition Post.. HJ JK KL LM. Substitution Prop.. HK KM. Substitution Prop.. H K K M. Def. of segments J H K L M. 3. For Questions and, use the figure at the right.. Find the measure of the numbered angles if m and m E 9. (Lesson -) E D 3.. If D is a median, D, and D., find. (Lesson -).. Write an inequalit to describe the possible values of. (Lesson -).. band of sequins that measures 0 inches is cut into two pieces so that their lengths are in a : ratio. Find the length of each piece. (Lesson -). Stan invests $ in a certificate of deposit that earns.% interest compounded annuall. Find the balance of his account after ears. (Lesson -).. For Questions 9 and 0, use the figure at the right. 9. Find QP to the nearest tenth. (Lesson -) P 0 M 0 30 L 9 Q R Find LM and PM. (Lesson -3) 0. Glencoe/McGraw-Hill 0 Glencoe Geometr

71 NME DTE PERIOD Standardized Test Practice (hapters ) SORE Part : Multiple hoice Instructions: Fill in the appropriate oval for the best answer.. If T bisects YT, T bisects TZ, m YT, and m T, find m TZ. (Lesson -).. 3. D. Y T Z. D. Which statement is alwas true? (Lesson -) E. If right QPR has sides q, p, and r, where r is the hpotenuse, then r p q. F. If E F H J, then EF HJ. G. If lines KL and VT are cut b a transversal, then K L V T. H. If D R and R H are congruent, then R bisects D H.. E F G H 3. The equation for PT is ( 3). Determine an equation for a line perpendicular to PT. (Lesson 3-).. 3. D.. ngle Y in XYZ measures 90. X Y and Y Z each measure meters. lassif XYZ. (Lesson -) E. acute and isosceles F. equiangular and equilateral G. right and scalene H. right and isosceles 3.. D E F G H ssessments. Two sides of a triangle measure inches and 9 inches. Determine which cannot be the perimeter of the triangle. (Lesson -). 9 in.. in.. 3 in. D. in.. STR, so?. (Lesson -) ST E. F. G. TR R S RS H. R S ST. The Petronas Towers in Kuala Lumpur, Malasia, are meters tall. woman who is. meters tall stands 0 meters from the base of one tower. Find the angle of elevation between the woman s hat and the top of the tower to the nearest tenth. (Lesson -) D... Which equation can be used to find? (Lesson -) E. sin 3 F. cos 3 G. H. cos 3 sin D E F G H D. E F G H Glencoe/McGraw-Hill Glencoe Geometr

72 NME DTE PERIOD Standardized Test Practice (continued) Part : Grid In Instructions: Enter our answer b writing each digit of the answer in a column bo and then shading in the appropriate oval that corresponds to that entr. 9. Find the measure of the smaller of two complementar angles whose measures differ b 3. (Lesson -) 0. How man countereamples are necessar to prove that a statement is false? (Lesson -3). Find so that m. (Lesson 3-). Find c to the nearest tenth. (Lesson -). c m / / / / / /... 3 / / Part 3: Short Response Instructions: Show our work or eplain in words how ou found our answer. 3. If DEF HJK, m D, m J 3, and m F 9, find. (Lesson -3) 3.. Use the Eterior ngle Inequalit Theorem to list all of the angles whose measures are less than m. (Lesson -) 3 D. For Questions and, use the figure at the right.. Determine whether EFH JGH. (Lesson -3) E 30 F G 0 H J.. If G is the midpoint of F H, find. (Lesson -3). Glencoe/McGraw-Hill Glencoe Geometr

73 NME DTE PERIOD Unit Review (hapter ) SORE. Use a protractor to classif UVW, UWX, and XWY as acute, equiangular, obtuse, or right. V W X Y. U. In the figure,. Find the measures of the numbered angles. D G E F H. 3. Name the corresponding congruent sides for FP STX. 3.. Determine whether PQR given (, ), (, 3), (, ), P(, ), Q(, 9), and R(, ).. In the figure, L K bisects JKM and KLJ KLM. Determine which theorem or postulate can be used to prove that JKL MKL. K L J M.. ssessments. Triangle is isosceles with. Name a pair of congruent angles in this triangle... Name the missing coordinates for isosceles right JKL with legs b units long. K(?,?). J(?,?) L(?,?) For Questions and 9, refer to the figure.. Find a and m ZWT if Z W is an altitude of XYZ, m ZWT 3a, and m TWY a 3. X W Z T Y. 9. Determine which angle has the greatest measure: YWZ, WZY, or ZYW Mr. Ramirez bought a stove and a dishwasher for just over $0. State the assumption ou would make to start an indirect proof to show that at least one of the appliances cost more than $ Glencoe/McGraw-Hill 3 Glencoe Geometr

74 NME DTE PERIOD Unit Review (continued). Determine whether feet, 3 feet, and feet can be the lengths of the sides of a triangle.. ase has a 3-inch television and a -inch television in her home. What is the ratio of the sizes of the smaller and larger TVs?.. 3. If EFG EJK, find, JK, KG, and the scale factor relating EFG to EJK. E 9 3 K J G 0 F ( ) 3.. Find... Find the perimeter of if XYZ. X Y 30 Z.. le has $0 in a bank account that earns.% interest. If the interest is compounded annuall and he does not make an withdrawals, find the balance of his account after 3 ears.. Find the geometric mean between and to the nearest tenth.. Determine whether, 0, and 3 are the measures of the sides of a right triangle. Then state whether the form a Pthagorean triple. 9. The diagonal of a square is centimeters long. Find the perimeter of the square to the nearest tenth. 0. Find m P to the nearest tenth in right MNP for M(3, ), N(3, ), and P(, ) For Questions and, refer to the figure. S. Find m S if m T, t, and s 33. t r.. Solve RST if t, s, and m R 0. R s T. Glencoe/McGraw-Hill Glencoe Geometr

75 NME DTE PERIOD First Semester Test (hapter ) SORE For Questions, write the letter for the correct answer in the blank at the right of each question.. ngles FH and HF form a linear pair and m FH 3. Find m HF D... Given (, ), D(, 0), and F(3, ), which of the following is a true conjecture?. DF is a right triangle.. DF is an isosceles triangle.. DF is an equilateral triangle. D., D, and F do not form a triangle. 3. Which is the inverse of the statement If, then 3?. If 3, then.. If, then 3.. If, then 3. D. If 3, then Find the slope of a line that is perpendicular to GH D. 3. Find the distance between parallel lines and m whose equations are 3 and 3 9. G O H. ssessments... 9 D. 9. Find sin P D. 0 P 0 Q R.. Find m G D.. F 3 H G.. Find c and PK if P is between L and K, LP c, PK c, and LK 3. Does P bisect L K? 9. Determine the distance between (, ), and ( 30, ) on a coordinate plane. State the coordinates of the midpoint of. Justif each statement with a propert or definition. 0. If D, then D.. If and 3 are complementar, then m m Glencoe/McGraw-Hill Glencoe Geometr

76 NME DTE PERIOD First Semester Test (continued). If the measures of two angles of a triangle are and 30, is the triangle acute, obtuse, or right? Eplain our reasoning.. 3. Identif the congruent triangles in the figure. J K P M 3. L N For Questions and, refer to the figure. Triangle is an isosceles right triangle. D. If D bisects, find m and m... Determine the coordinates of,, and, if the triangle has legs n units long.. For Questions, refer to the figure.. Write a statement using,, or to describe the measures of D and D. F 0 D 3 E.. Write an inequalit to represent the possible measures of D E.. If m F 3 and m D 3, write an inequalit to describe the possible values of Identif the similar triangles, find MN, and state the scale factor from the smaller triangle to the larger triangle. L J 3 N K M Find the first three iterates of ( 3) if initiall equals 0.. plane is fling at 3,000 feet, and the pilot wants to descend to,000 feet over the net 0 miles. What should be his angle of depression to the nearest tenth? (Hint: 0 feet mile). Solve DEF if DE, EF, and m E 9. Round angle measures to the nearest degree and side measures to the nearest tenth Glencoe/McGraw-Hill Glencoe Geometr

77 Standardized Test Practice Student Record Sheet (Use with pages of the Student Edition.) Glencoe/McGraw-Hill Glencoe Geometr nswers Select the best answer from the choices given and fill in the corresponding oval. 3 D D D D D D D NME DTE PERIOD Part Multiple hoice Part Multiple hoice Part Short Response/Grid In Part Short Response/Grid In Part 3 Open-Ended Part 3 Open-Ended Solve the problem and write our answer in the blank. For Questions, 9,, and, also enter our answer b writing each number or smbol in a bo. Then fill in the corresponding oval for that number or smbol. (grid in) 9 9 (grid in) 0 (grid in) (grid in) / / / / / / / / Record our answers for Question 3 on the back of this paper.