Geometry Unit 7 - Notes Right Triangles and Trigonometry

Transcription

1 Geometry Unit 7 - Notes Right Triangles and Trigonometry Review terms: 1) right angle ) right triangle 3) adjacent 4) Triangle Inequality Theorem Review topic: Geometric mean a = = d a d Syllabus Objective: The student will eplore right triangles and their relationships. Right Triangle cute Leg Hypotenuse Right cute Leg Pythagorean Triples a group of three whole numbers that satisfies the equation a + b = c, where c is the greatest number. It is important that students learn the and the triples and their multiples, they are used etensively on standardized tests! Especially those tests that do not allow calculators. Eamples: Given a 6, 8, 10 right triangle. Provide three multiples that result in right triangles and identify the scale factor (s.f.). a) 3, 4, 5 b) 1, 16, 0 c) 30, 40, 50 6 ( 1 ), 8( 1 ), 10( 1 ) 6( 1 ), 8( 1 ), 10( 1 ) 6( 5 1 ), 8( 5 1 ), 10( 5 1 ) Solutions: a) s.f. = 1 b) s.f. = 1 c) s.f. = 5 1 Unit 7 Right Triangles and Trigonometry Page 1 of 19

2 Syllabus Objective: 7. - The student will eplore geometric mean relationships within a right triangle. Theorem: If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. {Whole ~ Small ~ Medium } egin with a right triangle. Have students construct an altitude from the right angle. Students should be able to name and identify the corresponding parts of the three similar triangles that are now in the diagram. Etend into the geometric mean of the altitude and a segment of the hypotenuse. Theorem: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths piece1 altitude of the two segments. { = altitude = piece piece } 1 altitude piece Theorem: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric men of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to hyp. leg1 or the leg. { = leg = hyp. adj. piece } 1or 1or leg adj. piece 1or 1or Eamples: a) Write a proportion using the altitude as the geometric mean. D piece1 altitude D D Proportion: = =. altitude piece D D Unit 7 Right Triangles and Trigonometry Page of 19

3 b) Write a proportion using a leg of the right triangle as the geometric mean. hyp. leg1 or Two possible proportions. = leg adj. piece 1or 1or D = OR =. D D c) Find the missing value. If N = 4 and N = 9, find N. 4 Let = N = 9 = = d) Find the missing value. If = 8 and N = 4, find N. Let = N. The hypotenuse would be (4 + ) = = 64 4 = 48 = Use this diagram for questions c & d. N **The Radius** The Problem: semi-circle is drawn to fit tightly inside a right-angled triangle with sides 3, 4 and 5, as shown: What is the horizontal gap between the rightmost point of the triangle and the end of the diameter near to it,? Unit 7 Right Triangles and Trigonometry Page 3 of 19

4 Solution: This is all about similar triangles Since O = OD (both are radii of the circle), O = OD (both are right angles), and the triangles share side O, so O OD. (HL) Therefore by PT, D = = 3. nd D = D = 5 3 =. ut OD. (~) OD 3 So = =. D 4 r 3 3 Now solving our proportion, = and r =, so E = r = 3, which leaves us with 4 = E = 4-3= 1. So the gap at the right of the triangle,, is equal to 1 unit. Syllabus Objective: The student will solve problems using the Pythagorean Theorem. Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Review the distance formula and how the Pythagorean Theorem can be derived from it. It is important that students understand what a, b, and c stand for in the Pythagorean Theorem. useful suggestion is to teach ( ) = ( ) + ( ) 1 hyp leg leg. egin with Pythagorean Triples when you start solving for side lengths using the Pythagorean Theorem. Eample: Use the Pythagorean Theorem to solve for the missing side length *This is a multiple of a 5, 1, 13. The scale factor is 1. = 6 Unit 7 Right Triangles and Trigonometry Page 4 of 19

5 Eample: You've just picked up a ground ball at first base, and you see the other team's player running towards third base. How far do you have to throw the ball to get it from first base to third base, and throw the runner out? Solution: You need to throw the ball 17.3 feet to get it from first base to third base. Eample: You're locked out of your house and the only open window is on the second floor, 5 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window? Solution: You need a ladder that's 7 feet long. Unit 7 Right Triangles and Trigonometry Page 5 of 19

6 **The Magic of Pythagoras in an Unknown Rectangle!** We all get used to using the Pythagorean Theorem with numerical inputs, and an unknown 3rd length to find. The nice thing is that we aren't told, and don't need to know, the sides of the triangles concerned! The Problem: P D Given a rectangular courtyard D, and a well at the point P such that P = 10, P = 5 and P = 11, then find PD. Solution: Put in lines through P parallel to and D. Take = ( ) = ( m + n) = ( + y m + n) D( + y ) P = ( m) 0, 0, 0,,,,, 0 and,. y n P m D We now have a large number of right-angled triangles to which we can apply the Pythagorean Theorem: ( ) ( ) ( ) ( ) P = + m, P = y + n, P = + n, PD = y + m. ( P) + ( P ) = ( + m ) + ( y + n ) use the associative property to regroup terms, ( n ) ( y m ) ( P ) ( PD ). = = +,, Unit 7 Right Triangles and Trigonometry Page 6 of 19

7 From this, nd, So, ( ) ( ) P P = y + n ( + n ) = y. ( ) ( ) PD P = y + m ( + m ) = y. ( PD ) ( P) = ( P ) ( P ) ( PD ) = ( P) + ( P ) ( P ) = = 196. Therefore, PD = 14.,,,, The surprise in this question, perhaps, is that we do not need to know anything about the size of the rectangle to be able to solve it. Syllabus Objective: The student will solve problems using the converse of the Pythagorean Theorem and related theorems for obtuse or acute triangles. onverse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. The longest side must be used as the c value. Eample: Use the onverse of the Pythagorean Theorem to determine if the given side lengths create a right triangle. a) 35, 1, 8 b) 30, 34, 16 c) 6, 7, > > = 13 35? ? not a 15 = = 1156 Solutions: a) yes; 3, 4, 5 family. b) yes; 8, 15, 17 family. c) no, will not make a! (35 is the hypotenuse) (34 is the hypotenuse) (13 needs to be 85 ) Unit 7 Right Triangles and Trigonometry Page 7 of 19

8 Theorem: If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. If c < a + b, then is acute. Theorem: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. If c > a + b, then is obtuse. Eamples: Determine whether each set of numbers can be the measures of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. Solutions a) 7, 15, 1 1? > 74 a) obtuse b) 10, 1, 3 because of Ineq. b) not a triangle c) 44, 46, 91 because of Ineq. c) not a triangle d) 4, 1, 14 14? > 160 d) obtuse e) 4., 6.4, ? < 58.6 e) acute f) 4.5, 0, ? = 40.5 f) right Syllabus Objective: The student will solve problems utilizing the ratios of the sides of special right triangles Triangle Theorem: In a triangle, the hypotenuse is times as long as each leg Triangle Theorem: In a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Deriving triangle relationships: {also called an Isosceles Right Triangle} hoose a length for the two congruent sides,, and for the hypotenuse, y. Use the Pythagorean Theorem: y = + 45 y y = + y =. y = y = y = 45 Legs =, Hypotenuse =. Unit 7 Right Triangles and Trigonometry Page 8 of 19

9 \ Deriving triangle relationships: Starting with an equilateral triangle, construct an altitude. ecause the altitude will bisect a side, choose an even length for 30 the sides of the equilateral triangle,. The altitude will create two triangles. y The shorter leg will have a length of (half of ) and the longer leg will have an unknown length, y. 60 Use the Pythagorean Theorem: ( ) = + y 4 = + y y = 3. 3 = y y = 3 y = 3 Hypotenuse =, Shorter Leg =, Longer Leg = 3. Remind students: NOT all right triangles will have these relationships, only and right triangles! fter being taught these relationships, they seem to think they will magically work for all right triangles. ny right triangles without these specific sets of angles will still have to be solved using the Pythagorean Theorem and trig functions. Eamples: Given the special right triangles, find the value of and y. a) b) 30 y y Solutions: a) = 7 b) = 11 y = 7 y = Unit 7 Right Triangles and Trigonometry Page 9 of 19

10 Syllabus Objective: The student will define and apply basic trigonometric ratios of sine, cosine, and tangent. Trig ratios are based on similar triangles and used in right triangles to find measurements. They can be used to find the missing measurements on a right triangle when only two other measurements are known. Review topics: 1) adjacent ) opposite Y Z It is an accepted practice to name the sides of a triangle using the lower case letter of the angle opposite the side. z y Side y is opposite angle Y. {across from} Sides y and z are adjacent to angle X. {net to} X Trigonometric ratio a ratio of the lengths of two sides of a right triangle. Trigonometric Ratios Let be a right triangle. The sine, the cosine, and the tangent of the acute angle defined as follows. side opposite sin = hypotenuse side adjacent cos = hypotenuse side opposite tan = side adjacent It is a good idea to introduce the trig functions with a trig table. Make sure students write the equation they will be using to solve the question. Then have students use the decimal approimations and algebra techniques to solve the equations and use the table to find the closest value. What a relief when they are taught the calculator keystrokes! are Unit 7 Right Triangles and Trigonometry Page 10 of 19

11 Eample: Use the table and complete: Tan 73? Solution: Step 1: Locate 73 in the angle column. Step : Go across to the tangent column. Step 3: Read the value: Tan ngle Sine osine Tangent Use SOHHTO to aide in students memory of the epressions for the basic trigonometric functions. Opp dj Opp Sin = os = Tan = Hyp Hyp dj Eample: Find the sine, cosine, and tangent of the acute angles of the triangle. Epress each value as a decimal and round to the hundredths Solutions: sin = 15 sin = sin = sin =.80 5 cos = 0 cos = cos = cos =.60 5 tan = 15 tan = tan = tan In the above eample: Point out that the sin = cos and cos = sin. Will that be true in every case? Yes. lso, tan is the reciprocal of tan. lways true? Yes. nother useful trig identity: cos θ + sin θ = 1. heck it out Remind students that when finding the angle measurements, they already know the ratio of the sides. They must use inverse functions (usually the nd button on a scientific calculator) in order to compute missing angle values. Unit 7 Right Triangles and Trigonometry Page 11 of 19

12 Syllabus Objective: The student will solve problems using trigonometric ratios. ngle of elevation the angle that your line of sight looking upward makes with a line drawn horizontally. ngle of depression the angle that your line of sight looking downward makes with a line drawn horizontally. Eamples: Find the unknown measurement. a) t a certain time, a pole 6 ft tall casts a 3 ft shadow. What is the angle of elevation of the sun? 6 ft 6 tan = = (an be found using a trig table or using the inverse function on a scientific calculator.) 3 ft b) person in a lighthouse m above sea level sights a buoy in the water. If the angle of depression to the buoy is 5, how far from the base of the lighthouse is the buoy? 5 P m The distance between the buoy and the lighthouse can be found in two ways. L Method 1 OR Method m PL = 5 m PL = 90 5 = 65 tan5 = tan65 = ( tan5 ) = = ( tan65 ) = tan5 (.1445) The buoy is about 47 m away. Unit 7 Right Triangles and Trigonometry Page 1 of 19

13 Solve a right triangle determine the measures of all si parts of the triangle, 3 sides and 3 angles. In order to solve a right triangle you must know either: Two side lengths. OR One side length and one acute angle measure. Eample: Solve the right triangle. Round angle measures to the nearest degree and side measures to the nearest tenth. G 36 J 1 H Use the Pythagorean Theorem to find GJ. GJ = GJ tang = 1 36 m G 30. m J = = 60. Eample: Solve the application problem. Lenora wants to build the bike ramp shown. Find the length of the surface of the ramp. The surface of the ramp is the hypotenuse of the right triangle. Since we also know the side opposite the 0 angle, we look for a ratio that uses hypotenuse and 10 ft 10 opposite. That would be the sine ratio. sin0 = h 0 10 h = sin0 h 9 ft. Unit 7 Right Triangles and Trigonometry Page 13 of 19

14 This unit is designed to follow the Nevada State Standards for Geometry, SD syllabus and benchmark calendar. It loosely correlates to hapter 9 of McDougal Littell Geometry 004, sections The following questions were taken from the nd semester common assessment practice and operational eams for and would apply to this unit. Multiple hoice # Practice Eam (08-09) Operational Eam (08-09) 36. Nan stands at the corner of the rectangular driveway shown below. Gina stands at the corner of the rectangular garden shown below. D D 1 ft 15 ft 8 ft How far must Nan walk diagonally across the driveway ( to )?. 7 ft. 14 ft. 35 ft D. 49 ft 37. bo is shown below. 0 ft How much shorter in feet is it to walk diagonally through the garden ( to ) instead of walking around its edge ( to and to )?. 5 ft. 10 ft. 15 ft D. 5 ft Use the dimensions given in the diagram below. 4 cm 4 in. 3 cm 6 in. What is?. 6 in.. 38 in in. D in. 8 in. 1 cm What is the length of the diagonal from to? cm. 155 cm. 5 cm D. 13 cm Unit 7 Right Triangles and Trigonometry Page 14 of 19

15 38. Use the dimensions given in the diagram below. Use the dimensions given in the diagram below What is the value of? D The three sides of a triangle are 3 centimeters, 5 centimeters, and 7 centimeters. What is the best description for this triangle?. acute triangle. equiangular triangle. obtuse triangle D. right triangle 40. jet is flying 7 miles above the ground. The pilot spots an airport as shown below. What is the value of? D. 8 The three sides of a triangle are 5 cm, 6 cm, and 10 cm. What is the best description for this triangle?. obtuse triangle. equiangular triangle. acute triangle D. right triangle seagull in a palm tree spots a hot dog on the beach. S 7 mi d 45 What is the distance d from the plane to the airport?. 7 mi. 7 3 mi. 7 mi D. 14 mi How far is the seagull from the hot dog?. 9 ft. 18 ft ft D. 36 ft 18 ft 60 H Unit 7 Right Triangles and Trigonometry Page 15 of 19

16 41. Use the dimensions given in the diagram below. Use the dimensions given in the diagram below. 4 y 16 What is the value of y? D In rectangle D, D = 1 and m D = 30. What is the length of the longer side of the rectangle? D Use the table and the dimensions given in the diagram below. y What is the value of y? D square has a diagonal length of 1 inches. What is the length in inches of a side?. 3 in.. 6 in.. 6 in. D. 1 in. Use the table and the dimensions given in the diagram below r r θ sin θ cos θ tan θ What is the value of r? D θ sin θ cos θ tan θ What is the value of r? D Unit 7 Right Triangles and Trigonometry Page 16 of 19

17 44. Use the dimensions given in the right triangle below. Use the dimensions given in the right triangle below What is the cosine of ? What is the tangent of ?. 1 9 D Use the table and the dimensions given in the diagram below D. 6 8 Use the table and the dimensions given in the diagram below. ngle of descent ngle of descent 3.4 mi 10 mi 10 miles θ sin θ cos θ tan θ miles θ sin θ cos θ tan θ What is the approimate angle of descent? D. 0 What is the airplane s approimate angle of descent? D. 50 Unit 7 Right Triangles and Trigonometry Page 17 of 19

18 46. Use the dimensions given in the diagram below. Use the dimensions given in the diagram below ft Which equation would be used to find the distance h from the hot air balloon to the ground?. h = 150 tan 53. h = 150sin h = tan D. h = sin Use the table and the dimensions given in the diagram below. h d 47 Which equation would be used to find the distance d from the hot air balloon to point on the ground?. d = 10 tan 47. d = 10sin d = tan D. d = sin Use the table and the dimensions given in the diagram below. d 18 ft 100 ft h θ sin θ cos θ tan θ θ sin θ cos θ tan θ What is the approimate length d of the kite string?. 56 ft. 00 ft. 168 ft D. 100 ft What is the approimate height h of the kite off the ground in feet?. 50 feet. 64 feet. 77 feet D. 10 feet Unit 7 Right Triangles and Trigonometry Page 18 of 19

19 Free Response Practice Eam (08-09). Find the length of the altitude of an isosceles triangle with verte angle 10 and base length of 30 centimeters. Give answer in simplified radical form. Operational Eam (08-09). The diagonal of a square divides it into two triangles. The diagonal has length 10 centimeters. Find the area of the square. Unit 7 Right Triangles and Trigonometry Page 19 of 19