Radicals and the 12.5 Using the Pythagorean Theorem

Transcription

1 Radical and the Pythagorean Theorem. Finding Square Root. The Pythagorean Theorem. Approximating Square Root. Simplifying Square Root.5 Uing the Pythagorean Theorem I m pretty ure that Pythagora wa a Greek. I aid Greek, not Geek. Leonardo da Vinci claimed that the human face i made up of golden ratio. Let ee if the ame i true of a cat face.

2 What You Learned Before Complete the number entence with <, >, or =. Here how I remember the quare root of. February i the nd month. It ha 8 day. Split 8 into and. Move the decimal to get.. Example..0 Example Becaue i greater than, Becaue i le than 000,. i greater than i le than So,. >.0. So, 0. < Example Find three decimal that make the number entence 5. > Any decimal le than 5. will make the entence true. true. Sample anwer: 0., 9.05, 8.5 Complete the number entence with <, >, or = π. Find three decimal that make the number entence true > (6.EE.) Example Evaluate 8 ( ) ( 5). Firt: Parenthee 8 ( ) ( 5) = 8 6 ( ) Second: Exponent = 6 6 ( ) Third: Multiplication and Diviion (from left to right) = + Fourth: Addition and Subtraction (from left to right) = 8 Evaluate the expreion ( ) ( (6 ) )

3 . Finding Square Root COMMON CORE STATE STANDARDS 8.EE. when you are given the area of the quare? How can you find the ide length of a quare When you multiply a number by itelf, you quare the number. Symbol for quaring i nd power. = = 6 quared i 6. To undo thi, take the quare root of the number. Symbol for quare root i a radical ign. 6 = = The quare root of 6 i. ACTIVITY: Finding Square Root Work with a partner. Ue a quare root ymbol to write the ide length of the quare. Then find the quare root. Check your anwer by multiplying. a. Sample: = = ft Area = ft The ide length of the quare i feet. Check 0 b. Area = 8 yd c. Area = cm d. Area = 6 mi e. Area =.89 in. f. Area =. m g. Area = ft 9 50 Chapter Radical and the Pythagorean Theorem

4 ACTIVITY: The Period of a Pendulum Work with a partner. The period of a pendulum i the time (in econd) it take the pendulum to wing back and forth. The period T i repreented by T =. L, where L i the length of the pendulum (in feet). L Copy and complete the table. Then graph the function. I the function linear? L T T 8 Period of a Pendulum 7 6 Period (econd) L Length (feet). IN YOUR OWN WORDS How can you find the ide length of a quare when you are given the area of the quare? Give an example. How can you check your anwer? Ue what you learned about finding quare root to complete Exercie 6 on page 5. Section. Finding Square Root 5

5 . Leon Leon Tutorial Key Vocabulary quare root, p. 5 perfect quare, p. 5 radical ign, p. 5 radicand, p. 5 Study Tip EXAMPLE Zero ha one quare root, which i 0. A quare root of a number i a number that when multiplied by itelf, equal the given number. Every poitive number ha a poitive and a negative quare root. A perfect quare i a number with integer a it quare root. Finding Square Root of a Perfect Square Find the two quare root of = 9 and ( 7) ( 7) = 9 So, the quare root of 9 are 7 and 7. The ymbol i called a radical ign. It i ued to repreent a quare root. The number under the radical ign i called the radicand. Poitive Square Root Negative Square Root Both Square Root ± 6 = 6 = ± 6 = ± EXAMPLE Finding Square Root Find the quare root(). a. 5 Becaue 5 = 5, 5 = 5 = 5. 5 repreent the poitive quare root. b. 9 6 Becaue ( ) = 9 6, 9 6 ( = ) =. 9 repreent the 6 negative quare root. c. ±.5 ±.5 repreent both the poitive and negative quare root. Becaue.5 =.5, ±.5 = ±.5 =.5 and.5. Exercie 7 6 Find the two quare root of the number Find the quare root().. 5. ± Chapter Radical and the Pythagorean Theorem

6 EXAMPLE Evaluating Expreion Involving Square Root Evaluate the expreion. a = 5(6) + 7 Evaluate the quare root. = Multiply. = 7 Add. b = + 9 Simplify. = + Evaluate the quare root. = Add. Squaring a poitive number and finding a quare root are invere operation. Ue thi relationhip to olve equation involving quare. EXAMPLE Real-Life Application The area of a crop circle i 5,6 quare feet. What i the radiu of the crop circle? Ue. for π. A = π r Write the formula for the area of a circle. 5,6.r Subtitute 5,6 for A and. for π.,00 = r Divide each ide by..,00 = r Take poitive quare root of each ide. 0 = r Simplify. The radiu of the crop circle i about 0 feet. Exercie 8 Evaluate the expreion ( 9 0 ) 0. The area of a circle i 86 quare feet. Write and olve an equation to find the radiu of the circle. Ue. for π. Section. Finding Square Root 5

7 . Exercie Help with Homework. VOCABULARY I 6 a perfect quare? Explain.. REASONING Can the quare of an integer be a negative number? Explain.. NUMBER SENSE Doe 56 repreent the poitive quare root of 56, the negative quare root of 56, or both? Explain. 9+(-6)= +(-)= +(-9)= 9+(-)= Find the ide length of the quare. Check your anwer by multiplying.. Area = cm 5. Area =.69 km 6. 5 Area = yd 6 Find the two quare root of the number Find the quare root() ± ± ERROR ANALYSIS Decribe and correct the error in finding the quare root. Evaluate the expreion ( 80 ± = 5 5 ). ( ) +. NOTEPAD The area of the bae of a quare notepad i 9 quare inche. What i the length of one ide of the bae of the notepad? 5. CRITICAL THINKING There are two quare root of 5. Why i there only one anwer for the radiu of the button? r 5 Chapter Radical and the Pythagorean Theorem A 5 mm

8 Copy and complete the tatement with <, >, or = SAILBOAT The area of a ail i 0 quare feet. The bae and the height of the ail are equal. What i the height of the ail (in feet)? 0. REASONING I the product of two perfect quare alway a perfect quare? Explain your reaoning. h. ENERGY The kinetic energy K (in joule) of a falling apple i repreented by K = v, where v i the peed of the apple (in meter per econd). How fat i the apple traveling when the kinetic energy i joule? b Area cm. WATCHES The area of the two watch face have a ratio of 6 : 5. a. What i the ratio of the radiu of the maller watch face to the radiu of the larger watch face? b. What i the radiu of the larger watch face?. WINDOW The cot C (in dollar) of making a quare window with a ide length of n inche i repreented by C = n A window cot $55. What i 5 the length (in feet) of the window?. The area of the triangle i repreented by the formula A = ( )( 7)( 0), where i equal to half the perimeter. What i the height of the triangle? 7 cm cm 0 cm Evaluate the expreion. (Skill Review Handbook) MULTIPLE CHOICE Which of the following decribe the triangle? (Section.) A Acute B Right C Obtue D Equiangular Section. Finding Square Root 55

9 . The Pythagorean Theorem COMMON CORE STATE STANDARDS 8.G.6 8.G.7 triangle related? Pythagora wa a Greek mathematician and philoopher who dicovered one of the mot famou rule in mathematic. In mathematic, a rule i called a theorem. So, the rule that Pythagora dicovered i called the Pythagorean Theorem. How are the length of the ide of a right Pythagora (c. 570 B.C. c. 90 B.C.) ACTIVITY: Dicovering the Pythagorean Theorem Work with a partner. a. On grid paper, draw any right triangle. Label the length of the two horter ide (the leg) a and b. c b. Label the length of the longet ide (the hypotenue) c. c b a a b c. Draw quare along each of the three ide. Label the area of the three quare a, b, and c. d. Cut out the three quare. Make eight copie of the right triangle and cut them out. Arrange the figure to form two identical larger quare. c b a e. What doe thi tell you about the relationhip among a, b, and c? 56 Chapter Radical and the Pythagorean Theorem

10 ACTIVITY: Finding the Length of the Hypotenue Work with a partner. Ue the reult of Activity to find the length of the hypotenue of each right triangle. a. c b. 0 c c. d. c 0.6 c 0.8 ACTIVITY: Finding the Length of a Leg Work with a partner. Ue the reult of Activity to find the length of the leg of each right triangle. a. b. a b. IN YOUR OWN WORDS How are the length of the ide of a right triangle related? Give an example uing whole number. Ue what you learned about the Pythagorean Theorem to complete Exercie 5 on page 50. Section. The Pythagorean Theorem 57

11 . Leon Leon Tutorial Key Vocabulary theorem, p. 56 leg, p. 58 hypotenue, p. 58 Pythagorean Theorem, p. 58 Study Tip In a right triangle, the leg are the horter ide and the hypotenue i alway the longet ide. Side of a Right Triangle The ide of a right triangle have pecial name. The leg are the two ide that form the right angle. leg, a leg, b hypotenue, c The hypotenue i the ide oppoite the right angle. The Pythagorean Theorem Word In any right triangle, the um of the quare of the length of the leg i equal to the quare of the length of the hypotenue. Algebra a + b = c EXAMPLE Finding the Length of a Hypotenue 5 m c m Find the length of the hypotenue of the triangle. a + b = c Write the Pythagorean Theorem. 5 + = c Subtitute 5 for a and for b. 5 + = c Evaluate power. 69 = c Add. 69 = c Take poitive quare root of each ide. = c Simplify. The length of the hypotenue i meter. Find the length of the hypotenue of the triangle... in. 0 c 8 ft in. 5 c 5 ft 58 Chapter Radical and the Pythagorean Theorem

12 a EXAMPLE.9 cm. cm Finding the Length of a Leg Find the miing length of the triangle. a + b = c Write the Pythagorean Theorem. a +. =.9 Subtitute. for b and.9 for c. a +. = 8. Evaluate power. a = Subtract. from each ide. a = Take poitive quare root of each ide. The length of the leg i centimeter. EXAMPLE Standardized Tet Practice Ranger Station W x 6 km Group A N Group B y km km E 8 km S Hiking Group A leave a ranger tation and hike 8 kilometer outh then 6 kilometer wet. Group B leave the tation and hike kilometer eat then kilometer north. Uing the figure, how far apart are the two group of hiker? A 5 km B 0 km C 5 km D km The ditance between the group i the um of the hypotenue, x and y. Ue the Pythagorean Theorem to find x and y. a + b = c Write the Pythagorean Theorem. a + b = c = x Subtitute. + = y = x Evaluate power = y 00 = x Add. 5 = y 0 = x Take poitive quare root of each ide. 5 = y The ditance between the group of hiker i = 5 kilometer. So, the correct anwer i C. Exercie 8 Find the miing length of the triangle... a yd 6 yd b 0. m 9.6 m 5. WHAT IF? In Example, Group A hike kilometer outh and 9 kilometer wet. How far apart are the hiker? Section. The Pythagorean Theorem 59

13 . Exercie Help with Homework. VOCABULARY In a right triangle, how can you tell which ide are the leg and which ide i the hypotenue?. DIFFERENT WORDS, SAME QUESTION Which i different? Find both anwer. Which ide i the hypotenue? Which ide i the longet? a c Which ide i a leg? Which ide i oppoite the right angle? b 9+(-6)= +(-)= +(-9)= 9+(-)= Find the miing length of the triangle.. 6 cm 0 cm. 0 km in. b km c a 0.6 in mm 7. b 5 mm 7. ft c 9.6 ft 8. a yd yd 9. ERROR ANALYSIS Decribe and correct the error in finding the miing length of the triangle. 7 ft 5 ft a + b = c = c 67 = c 67 = c 5.6 ft c 0. TREE SUPPORT How long i the wire that upport the tree?. ft 50 Chapter Radical and the Pythagorean Theorem

14 Find the value of x.. 0 cm. mm 5 mm x. x 0 ft cm x 5 mm 6 ft in. d. FLAT SCREEN Televiion are advertied by the length of their diagonal. A tore ha a ale on televiion 0 inche and larger. I the televiion on ale? Explain. in. 5. BUTTERFLY Approximate the wingpan of the butterfly. Wingpan Hole Par 8 Yard Hole 5.8 cm cm 80 yd x 6. GOLF The figure how the location of a golf ball after a tee hot. How many feet from the hole i the ball? Tee 7. SNOWBALLS You and a friend tand back-to-back. You run 0 feet forward then 5 feet to your right. At the ame time, your friend run 6 feet forward then feet to her right. She top and hit you with a nowball. a. Draw the ituation in a coordinate plane. b. How far doe your friend throw the nowball? 8. The leg of a right triangle have length of 8 meter and meter. The hypotenue ha a length of 5x meter. What i the value of x? Find the quare root(). (Section.) 9. ± MULTIPLE CHOICE Which type of triangle can have an obtue angle? (Section.) A equiangular B right C iocele D equilateral Section. The Pythagorean Theorem 5

15 Study Help Graphic Organizer You can ue a ummary triangle to explain a topic. Here i an example of a ummary triangle for finding the length of the hypotenue of a triangle. Finding the length of the hypotenue of a triangle Pythagorean Theorem In any right triangle, the um of the quare of the length of the leg i equal to the quare of the length of the hypotenue. a c a + b = c Example : 6 ft b c 8 ft = c = c 00 = c 0 = c Make a ummary triangle to help you tudy thee topic.. finding quare root. evaluating expreion involving quare root. finding the length of a leg of a right triangle After you complete thi chapter, make ummary triangle for the following topic.. approximating quare root 5. implifying quare root What do you call a cheee ummary triangle that in t your? 5 Chapter Radical and the Pythagorean Theorem

16 .. Quiz Progre Check Find the two quare root of the number. (Section.) Find the quare root(). (Section.) ± 6.5 Evaluate the expreion. (Section.) Find the miing length of the triangle. (Section.) ft c a 0 ft 5 in. 5 in...6 cm 6.5 cm. 0 yd c b 5 yd. POOL The area of a circular pool cover i quare feet. Write and olve an equation to find the diameter of the pool cover. Ue. for π. (Section.) 5. LAND A quare parcel of land ha an area of million quare feet. What i the length of one ide of the parcel? (Section.) 6. FABRIC You are cutting a rectangular piece of fabric in half along the diagonal. The fabric meaure 8 inche wide and yard long. What i the length (in inche) of the diagonal? (Section.) Section.. Quiz 5

17 . Approximating Square Root COMMON CORE STATE STANDARDS 8.EE. 8.NS. 8.NS. quare root that are irrational? How can you find decimal approximation of You already know that a rational number i a number that can be written a the ratio of two integer. Number that cannot be written a the ratio of two integer are called irrational. Real Number Rational Integer Irrational Natural π π.5 0. ACTIVITY: Approximating Square Root Work with a partner. Archimede wa a Greek mathematician, phyicit, engineer, inventor, and atronomer. a. Archimede tried to find a rational number whoe quare i. Here are two that he tried and Are either of thee number equal to? How can you tell? b. Ue a calculator with a quare root key to approximate. Write the number on a piece of paper. Then enter it into the calculator and quare it. Then ubtract. Do you get 0? Explain. c. Calculator did not exit in the time of Archimede. How do you think he might have approximated? Archimede (c. 87 B.C. c. B.C.) Square Root Key 5 Chapter Radical and the Pythagorean Theorem

18 ACTIVITY: Approximating Square Root Geometrically Work with a partner. a. Ue grid paper and the given cale to draw a horizontal line egment unit in length. Label thi egment AC. b. Draw a vertical line egment unit in length. Label thi egment DC. c. Set the point of a compa on A. Set the compa to unit. Swing the compa to interect egment DC. Label thi interection a B. d. Ue the Pythagorean Theorem to how that the length of egment BC i unit. e. Ue the grid paper to approximate. D B Scale: of a unit 0 C A. Repeat Activity for a triangle in which egment CA i unit and egment BA i unit. Ue the Pythagorean Theorem to how that egment BC i 5 unit. Ue the grid paper to approximate 5.. IN YOUR OWN WORDS How can you find decimal approximation of quare root that are irrational? Ue what you learned about approximating quare root to complete Exercie 5 8 on page 59. Section. Approximating Square Root 55

19 . Leon Leon Tutorial Key Vocabulary irrational number, p. 56 real number, p. 56 A rational number i a number that can be written a the ratio of two integer. An irrational number cannot be written a the ratio of two integer. The quare root of any whole number that i not a perfect quare i irrational. The decimal form of an irrational number neither terminate nor repeat. Remember Decimal that terminate or repeat are rational. Real Number Rational number and irrational number together form the et of real number. Real Number Rational Integer Irrational Natural π π.5 0. EXAMPLE Claifying Real Number Tell whether the number i rational or irrational. Explain. a. b. c. d. Number Rational or Irrational Reaoning Irrational i not a perfect quare. 0.6 Rational 0.6 i a repeating decimal. 7 0 Rational can be written a Rational 0.85 can be written a 7 0. Tell whether the number i rational or irrational. Explain. Exercie Chapter Radical and the Pythagorean Theorem

20 EXAMPLE Approximating Square Root Etimate 5 to the nearet integer. Ue a number line and the quare root of the perfect quare nearet to the radicand. The nearet perfect quare le than 5 i 9. The nearet perfect quare greater than 5 i 6. Graph 5. 9 = 7 6 = 8 Becaue 5 i cloer to 9 than to 6, 5 i cloer to 7 than to 8. So, 5 7. Exercie 8 Etimate to the nearet integer EXAMPLE Comparing Real Number a. Which i greater, 5 or? Graph the number on a number line. 5 =.75 = 9 = i to the right of 5. So, b. Which i greater, 0. 6 or 0.6? Graph the number on a number line. i greater. 0.6 = i to the right of 0.6. So, 0. 6 i greater. Which number i greater? Explain. Exercie , 9. 0, 5 0., Section. Approximating Square Root 57

21 EXAMPLE Approximating an Expreion The radiu of a circle with area A i approximately A. The area of a circular moue pad i 5 quare inche. Etimate it radiu. A = 5 Subtitute 5 for A. = 7 Divide. The nearet perfect quare le than 7 i 6. The nearet perfect quare greater than 7 i = 5 = 5 Becaue 7 i cloer to 6 than to 5, 7 i cloer to than to 5. The radiu i about inche.. WHAT IF? The area of a circular moue pad i 6 quare inche. Etimate it radiu. EXAMPLE 5 Real-Life Application The ditance (in nautical mile) you can ee with a pericope i.7 h, where h i the height of the pericope above the water. Can a pericope that i 6 feet above the water ee twice a far a a pericope that i feet above the water? Explain. h Ue a calculator to find the ditance. feet above water 6 feet above water.7 h =.7 Subtitute for h..7 h = Ue a calculator..87 You can ee.87. time farther with the pericope that i 6 feet.0 above the water than with the pericope that i feet above the water. No, the pericope that i 6 feet above the water cannot ee twice a far.. You ue a pericope that i 0 feet above the water. Can you ee farther than nautical mile? Explain. 58 Chapter Radical and the Pythagorean Theorem

22 . Exercie Help with Homework. VOCABULARY What i the difference between a rational number and an irrational number?. WRITING Decribe a method of approximating.. VOCABULARY What are real number? Give three example.. WHICH ONE DOESN T BELONG? Which number doe not belong with the other three? Explain your reaoning (-6)= +(-)= +(-9)= 9+(-)= Tell whether the rational number i a reaonable approximation of the quare root , , , , 5 Tell whether the number i rational or irrational. Explain π ERROR ANALYSIS Decribe and correct the error in claifying the number. i irrational. 6. SCRAPBOOKING You cut a picture into a right triangle for your crapbook. The length of the leg of the triangle are inche and 6 inche. I the length of the hypotenue a rational number? Explain. Rational Integer Natural Real Number Irrational 7. VENN DIAGRAM Place each number in the correct area of the Venn Diagram. a. Your age b. The quare root of any prime number c. The ratio of the circumference of a circle to it diameter Section. Approximating Square Root 59

23 Etimate to the nearet integer CHECKERS A checkerboard i 8 quare long and 8 quare wide. The area of each quare i quare centimeter. Etimate the perimeter of the checkerboard. Which number i greater? Explain. 5. 0, ,.5 7., 0 8., , , 9 8. FOUR SQUARE The area of a four quare court i 66 quare feet. Etimate the length of one of the ide of the court.. RADIO SIGNAL The maximum ditance (in nautical mile) that a radio tranmitter ignal can be ent i repreented by the expreion. h, where h i the height (in feet) above the tranmitter. Etimate the maximum ditance x (in nautical mile) between the plane that i receiving the ignal and the tranmitter. Round your anwer to the nearet tenth. x,000 ft Not drawn to cale. OPEN-ENDED Find two number a and b that atify the diagram. 9 a b 0 50 Chapter Radical and the Pythagorean Theorem

24 Etimate to the nearet tenth r 6.76 m 7. ROLLER COASTER The velocity v (in meter per econd) of a roller coater i repreented by the equation v = 6r, where r i the radiu of the loop. Etimate the velocity of a car going around the loop. Round your anwer to the nearet tenth. 8. I a rational number? I a rational number? Explain WATER BALLOON The time t (in econd) it take a water balloon to fall d meter i repreented by the equation d t =. Etimate the time it take the balloon to fall.9 to the ground from a window that i meter above the ground. Round your anwer to the nearet tenth. 0. Determine if the tatement i ometime, alway, or never true. Explain your reaoning and give an example of each. a. A rational number multiplied by a rational number i rational. b. A rational number multiplied by an irrational number i rational. c. An irrational number multiplied by an irrational number i rational. Simplify the expreion. (Skill Review Handbook). x + y 5x. π + 8(t π) t. 7k 9 + k. MULTIPLE CHOICE What i the ratio (red to blue) of the correponding ide length of the imilar triangle? (Section 5.) A : B 5: C : D :5 Section. Approximating Square Root 5

25 . Simplifying Square Root COMMON CORE STATE STANDARDS 8.NS. golden ratio? Two quantitie are in the golden ratio if the ratio between the um of the quantitie and the greater quantity i the ame a the ratio between the greater quantity and the leer quantity. x + x = x How can you ue a quare root to decribe the x x + In a future algebra coure, you will be able to prove that the golden ratio i + 5 Golden ratio. ACTIVITY: Contructing a Golden Ratio Work with a partner. a. Ue grid paper and the given cale to draw a quare that i unit by unit (blue). F E b. Draw a line from midpoint C of one ide of the quare to the oppoite corner D, a hown. c. Ue the Pythagorean Theorem to find the length of egment CD. D 5 d. Set the point of a compa on C. Set the compa radiu to the length of egment CD. Swing the compa to interect line BC at point E. C e. The rectangle ABEF i called a golden rectangle becaue the ratio of it ide length i the golden ratio. A Scale: of a unit 0 B f. Ue a calculator to find a decimal approximation of the golden ratio. Round your anwer to two decimal place. 5 Chapter Radical and the Pythagorean Theorem

26 ACTIVITY: The Golden Ratio and the Human Body Work with a partner. Leonardo da Vinci wa one of the firt to notice that there are everal ratio in the human body that approximate the golden ratio. a. Ue a tape meaure or two yardtick to meaure the length hown in the diagram for both you and your partner. (Take your hoe off before meauring.) f e g c b Navel b. Copy the table below. Record your reult in the firt two column. h c. Calculate the ratio hown in the table. a d. Leonardo da Vinci tated that for many people, the ratio are cloe to the golden ratio. How cloe are your ratio? d You Partner a = b = a b = a = b = a b = c = d = c d = c = d = c d = e = f = e f = e = f = e f = g = h = g h = g = h = g h =. IN YOUR OWN WORDS How can you ue a quare root to decribe the golden ratio? Ue the Internet or ome other reference to find example of the golden ratio in art and architecture. Ue what you learned about quare root to complete Exercie 5 on page 56. Section. Simplifying Square Root 5

27 . Leon Leon Tutorial You can add or ubtract radical expreion the ame way you combine like term, uch a 5x + x = 9x. Reading EXAMPLE Do not aume that radical that have different radicand cannot be implified. An expreion uch a + can eaily be implified. Adding and Subtracting Square Root a. Simplify = (5 + ) Ue the Ditributive Property. = 9 Simplify. b. Simplify 7. 7 = ( 7) Ue the Ditributive Property. = 5 Simplify. Exercie 6 Simplify the expreion To implify quare root that are not perfect quare, ue the following property. Product Property of Square Root Algebra xy = x y, where x, y 0 Number = = Study Tip EXAMPLE A quare root i implified when the radicand ha no perfect quare factor other than. Exercie 6 0 Simplifying Square Root Simplify = 5 Factor uing the greatet perfect quare factor. = 5 Ue the Product Property of Square Root. = 5 Simplify. Simplify the expreion Chapter Radical and the Pythagorean Theorem

28 Quotient Property of Square Root x x Algebra y = y, where x 0 and y > 0 Number 7 9 = 9 = 7 7 EXAMPLE Simplifying Square Root Simplify 6. 6 = 6 = Ue the Quotient Property of Square Root. Simplify. Remember EXAMPLE The volume V of a rectangular prim i the product of the area of it bae B and it height h. V = B h Finding a Volume Find the volume of the rectangular prim. m 0 m 5 m V = B h Write formula for volume. = ( 5 ) ( 0 ) ( ) Subtitute. = 5 0 Ue the Product Property of Square Root. = 00 Multiply. = 0 Simplify. The volume i 0 cubic meter. Simplify the expreion. Exercie b 0. WHAT IF? In Example, the height of the rectangular prim i 8 meter. Find the volume of the prim. Section. Simplifying Square Root 55

29 . Exercie Help with Homework. WRITING Decribe how combining like term i imilar to adding and ubtracting quare root.. WRITING How are the Product Property of Square Root and the Quotient Property of Square Root imilar? 9+(-6)= +(-)= +(-9)= 9+(-)= Find the ratio of the ide length. I the ratio cloe to the golden ratio? ft yd 50 m 6 ft Simplify the expreion. yd 5 m ERROR ANALYSIS Decribe and correct the error in implifying the expreion = 7 0 Simplify the expreion c 5. RAIN GUTTER A rain gutter i made from a ingle heet of metal. What i the length of the red cro-ection? in. in. in. 56 Chapter Radical and the Pythagorean Theorem

30 Simplify the expreion VOLUME What i the volume of the aquarium (in cubic feet)? 0. RATIO The ratio : x i equivalent to the ratio x : 5. What are the poible value of x? ft 0 ft 5 ft 0 ft ft. BILLBOARD The billboard ha the hape of a rectangle. a. What i the perimeter of the billboard? b. What i the area of the billboard?. MT. FUJI Mt. Fuji i in the hape of a cone with a volume of about 75π cubic kilometer. What i the radiu of the bae of Mt. Fuji?. A block of ice i in the hape of a quare prim. You want to put the block of ice in a cylindrical cooler. The equation = r repreent the minimum radiu r needed for The height of Mt. Fuji i.8 kilometer. the block of ice with ide length to fit in the cooler. a. Solve the equation for r. r b. Ue the equation in part (a) to find the minimum radiu needed when the ide length of the block of ice i 98 inche. Find the miing length of the triangle. (Section.). m c 5. 0 in. 6 in. b 6. cm a m 5 cm 7. MULTIPLE CHOICE Where i 0 on a number line? (Section.) A Between 9 and 0 B Between 9 and 0 C Between 0 and D Between 0 and Section. Simplifying Square Root 57

31 .b Cube Root Leon Tutorial A cube root of a number i a number that when multiplied by itelf, and then multiplied by itelf again, equal the given number. A perfect cube i a number that can be written a the cube of an integer. The ymbol i ued to repreent a cube root. Remember EXAMPLE The ymbol for cubing i the third power. So, =. Finding Cube Root Find the cube root. a. 8 = 8 Becaue = 8, 8 = =. b. 7 ( ) ( ) = 7 Becaue ( ) = 7, 7 = ( ) =. EXAMPLE Simplifying Expreion Involving Cube Root Simplify the expreion. a = (9 5) Ue the Ditributive Property. = Simplify. b = 8(0) + Evaluate the cube root. = 80 + Multiply. = 8 Add. Find the cube root Simplify the expreion A Chapter Radical and the Pythagorean Theorem

32 EXAMPLE Comparing Real Number Which i greater, 50 or.? Graph the number on a number line. To graph 50, ue the cube root of the perfect cube nearet to the radicand. The nearet perfect cube le than 50 i 7. The nearet perfect cube greater than 50 i i to the right of.. So, 50 i greater. Cubing a number and finding a cube root are invere operation. Ue thi relationhip to olve equation involving cube. EXAMPLE Real-Life Application Remember The volume V of a cube with ide length i given by V =. The urface area S i given by S = 6. Find the urface area of the baeball diplay cae. Ue the formula for the volume of a cube to find the ide length. V = Write formula for volume. 5 = Subtitute 5 for V. 5 = Take the cube root of each ide. 5 = Simplify. The ide length i 5 inche. Ue a formula to find the urface area of the cube. S = 6 Write formula for urface area. = 6(5) Subtitute 5 for. = 6(5) Evaluate 5. = 50 Simplify. Volume 5 in. The urface area of the baeball diplay i 50 quare inche. Which number i greater? Explain. 5..9, , 7. 00, , 9., , 60. The volume of a cube i 5 cubic centimeter. Find the urface area of the cube. Cube Root 57B

33 .5 Uing the Pythagorean Theorem COMMON CORE STATE STANDARDS 8.G.6 8.G.7 8.G.8 How can you ue the Pythagorean Theorem to olve real-life problem? ACTIVITY: Uing the Pythagorean Theorem Work with a partner. a. A baeball player throw a ball from econd bae to home plate. How far doe the player throw the ball? Include a diagram howing how you got your anwer. Decide how many decimal point of accuracy are reaonable. Explain your reaoning. b. The ditance from the pitcher mound to home plate i 60.5 feet. Doe thi form a right triangle with firt bae? Explain your reaoning. 90 ft 90 ft ACTIVITY: Firefighting and Ladder Work with a partner. The recommended angle for a firefighting ladder i 75. When a 0-foot ladder i put up againt a building at thi angle, the bae of the ladder i about 8 feet from the building. The bae of the ladder i 8 feet above the ground. How high on the building will the ladder reach? Round your anwer to the nearet tenth. 0 ft x 8 ft 8 ft 58 Chapter Radical and the Pythagorean Theorem

34 ACTIVITY: Finding Perimeter Work with a partner. Find the perimeter of each figure. Round your anwer to the nearet tenth. Did you ue the Pythagorean Theorem? If o, explain. a. Right triangle b. Trapezoid c. Parallelogram in. cm cm 6 ft in. cm in. ft ACTIVITY: Writing a Formula Work with a partner. a. Write a formula for the area of an equilateral triangle with ide length. b. Ue your formula to find the area of an equilateral triangle with a ide length of 0 inche. 5. IN YOUR OWN WORDS How can you ue the Pythagorean Theorem to olve real-life problem? 6. Decribe a ituation in which you could ue the Pythagorean Theorem to help make deciion. Give an example of a real-life problem. Ue what you learned about uing the Pythagorean Theorem to complete Exercie 5 on page 5. Section.5 Uing the Pythagorean Theorem 59

35 .5 Leon Leon Tutorial EXAMPLE Finding a Ditance in a Coordinate Plane The park i 5 mile eat of your home. The library i mile north of the park. How far i your home from the library? Round your anwer to the nearet tenth. Key Vocabulary Pythagorean triple, p. 5 Plot a point for your home at the origin in a coordinate plane. Then plot point for the location of the park and the library to form a right triangle. 6 N 5 Library a + b = c Write the Pythagorean Theorem. + 5 = c Subtitute for a and 5 for b = c Evaluate power. = c = c 6. c Home W c 5 Park E S Add. Take poitive quare root of each ide. Ue a calculator. Your home i about 6. mile from the library.. The pot office i mile wet of your home. Your chool i mile north of the pot office. How far i your home from your chool? Round your anwer to the nearet tenth. Exercie 6 8 EXAMPLE Real-Life Application Find the height of the firework. Round your anwer to the nearet tenth. a + b = c x + 00 = 5 5 m x x + 90,000 =,5 x =,5 00 m.5 m x =,5 Not drawn to cale x 9. Write the Pythagorean Theorem. Subtitute. Evaluate power. Subtract 90,000 from each ide. Take poitive quare root of each ide. Ue a calculator. The height of the firework i about = 50.6 meter. 50 Chapter MSCC7Ape_05.indd 50 Radical and the Pythagorean Theorem // :9:0 PM

36 Exercie 9. WHAT IF? In Example, the ditance between you and the firework i 50 meter. Find the height of the firework. Round your anwer to the nearet tenth. A Pythagorean triple i a et of three poitive integer a, b, and c where a + b = c. Convere of the Pythagorean Theorem If the equation a + b = c i true for the ide length of a triangle, then the triangle i a right triangle. When uing the convere of the Pythagorean Theorem, alway ubtitute the length of the longet ide for c. a b c EXAMPLE Identifying a Right Triangle Tell whether the given triangle i a right triangle. a. 9 cm cm b. 8 ft ft 0 cm ft a + b = c a + b = c =? + 8 =? =? 68 + =? = It i a right triangle. It i not a right triangle. Exercie 8 Tell whether the triangle with the given ide length i a right triangle.. 5 m. 6 m 7 m 8 in. in. 0 in. 5. yd, yd, yd 6..5 mm, mm, 0.75 mm Section.5 Uing the Pythagorean Theorem 5

37 .5 Exercie Help with Homework. WRITING How can the Pythagorean Theorem be ued to find ditance in a coordinate plane?. WHICH ONE DOESN T BELONG? Which et of number doe not belong with the other three? Explain your reaoning., 6, 8 6, 8, 0 5,, 7,, 5 9+(-6)= +(-)= +(-9)= 9+(-)= Find the perimeter of the figure. Round your anwer to the nearet tenth.. Right triangle. Parallelogram 5. Square 6 m 6 ft yd 0 m ft 9 ft yd Find the ditance d. Round your anwer to the nearet tenth y d y d y d x x x Find the height x. Round your anwer to the nearet tenth ft x 60 yd x 5 m x ft 50 yd 6 m. m. BICYCLE You ride your bicycle along the outer edge of a park. Then you take a hortcut back to where you tarted. Find the length of the hortcut. Round your anwer to the nearet tenth. hortcut 00 m 60 m 5 Chapter Radical and the Pythagorean Theorem

38 Tell whether the triangle with the given ide length i a right triangle.. 7 in.. 0 km 5. 5 km 8 in. 5 in km.5 ft 8 ft 8.5 ft 6. mm, 9 mm, mm mi, 5 mi, mi 8.. m,.8 m, 5 m 9. STAIRS There are tep in the taircae. Find the ditance from point A to point B (in feet). Round your anwer to the nearet tenth. 8 in. 0 in. B A 0. AIRPORT Which plane i cloer to the tower? Explain. Airport Altitude: 0,000 ft Plane A Altitude: 8000 ft Plane B 5 km km Not drawn to cale. PROJECT Find a hoebox or ome other mall box. A a. Meaure the dimenion of the box. Height b. Without meauring, find length BC and length AB. c. Ue a piece of tring and a ruler to check the length you found in part (b). C Length B Width. Plot the point (, ), (, ), and (, 6) in a coordinate plane. Are the point the vertice of a right triangle? Explain. Find the mean, median, and mode of the data. (Skill Review Handbook)., 9, 7, 5,,.,, 6, 7,, 9, , 59,, 7, MULTIPLE CHOICE What i the um of the angle meaure of an octagon? (Section.) A 70 B 080 C 0 D 800 Section.5 Uing the Pythagorean Theorem 5

39 ..5 Quiz Progre Check Tell whether the number i rational or irrational. Explain. (Section.) Etimate to the nearet integer. (Section.) Which number i greater? Explain. (Section.) 7., 5 8..,. 8 Simplify the expreion. (Section.) Find the volume of the rectangular prim. (Section.).. 5 in. 0.6 cm 5 in. 5 in. cm 0. cm Ue the figure to anwer Exercie 7. Round your anwer to the nearet tenth. (Section.5). How far i the cabin from the peak? y Fire Tower 5. How far i the fire tower from the lake? 6. How far i the lake from the peak? 7. You are tanding at ( 5, 6). How far are you from the lake? Cabin Bae Camp Lake 5 x 5 Peak unit = km Tell whether the triangle with the given ide length i a right triangle. (Section.5) 8. 6 ft 8 ft 9..5 m. m.7 m 5 ft 5 Chapter Radical and the Pythagorean Theorem

40 Chapter Review Review Key Vocabulary quare root, p. 5 perfect quare, p. 5 radical ign, p. 5 radicand, p. 5 theorem, p. 56 leg, p. 58 hypotenue, p. 58 Pythagorean Theorem, p. 58 Vocabulary Help irrational number, p. 56 real number, p. 56 Pythagorean triple, p. 5 Review Example and Exercie. Finding Square Root (pp ) Find the quare root(). a. 6 6 repreent the negative quare root. Becaue 6 = 6, 6 = 6 = 6. b repreent the poitive quare root. Becaue. =.96,.96 =. =.. c. ± 6 8 Becaue ( 9) = 6 8, ± 6 8 ( = ± 9) = ± 6 repreent both the poitive 8 and negative quare root. 9 and 9. Find the two quare root of the number Find the quare root() Evaluate the expreion ± ( 8 ) Chapter Review 55

41 . The Pythagorean Theorem (pp. 56 5) Find the length of the hypotenue of the triangle. a + b = c 7 + = c Subtitute. Write the Pythagorean Theorem = c Evaluate power. 65 = c Add. 65 = c 5 = c Simplify. Take poitive quare root of each ide. The length of the hypotenue i 5 yard. yd c 7 yd Find the miing length of the triangle. 0. in. c. b 0. cm 5 in. 0.5 cm. Approximating Square Root (pp. 5 5) Etimate to the nearet integer. Ue a number line and the quare root of the perfect quare nearet to the radicand. The nearet perfect quare le than i 5. The nearet perfect quare greater than i 6. Graph Becaue i cloer to 6 than to 5, i cloer to 6 than to 5. So, 6. Etimate to the nearet integer Chapter Radical and the Pythagorean Theorem

42 . Simplifying Square Root (pp. 5 57) Simplify 8. 8 = 7 Simplify 6. Factor uing the greatet perfect quare factor. = 7 Ue the Product Property of Square Root. = 7 Simplify. 6 = 6 = 8 Ue the Quotient Property of Square Root. Simplify. Simplify the expreion Uing the Pythagorean Theorem (pp. 58 5) Find the height of the tilt walker. Round your anwer to the nearet tenth. a + b = c Write the Pythagorean Theorem. 6 + x = Subtitute. 6 + x = 69 Evaluate power. x = Subtract 6 from each ide. x = Take poitive quare root of each ide. ft x x.5 Ue a calculator. The height of the tilt walker i about.5 feet. 6 ft Find the height x. Round your anwer to the nearet tenth, if neceary ft x 77 ft ft x ft Chapter Review 57

43 Chapter Tet Tet Practice Find the quare root() ± 00 9 Evaluate the expreion Find the miing length of the triangle. a 6 in. in. Tell whether the number i rational or irrational. Explain. 7. 6π 8. 9 Which number i greater? Explain , Simplify the expreion. 0. 5, ft x. Tell whether the triangle i a right triangle. 80 mm 9 mm 89 mm. ROBOT Find the height of the 5. SUPERHERO Find dinoaur robot. the altitude of the uperhero balloon. 5 m x ft 6 ft 7 m 58 Chapter Radical and the Pythagorean Theorem

44 Standardized Tet Practice. The period T of a pendulum i the time, in econd, it take the pendulum to wing back and forth. The period can be found uing the formula T =. L, where L i the length, in feet, of the pendulum. A pendulum ha a length of feet. Find it period. (8.EE.) Tet-Taking Strategy Anwer Eay Quetion Firt A. 5. ec C.. ec B.. ec D.. ec. The tep Pat took to write the equation in lope-intercept form are hown below. What hould Pat change in order to correctly rewrite the equation in lope-intercept form? (8.EE.6) Scan the tet and anwer the eay quetion firt. You know the quare root of i. x 6y = x = 6y + x = y + F. Ue the formula m = rie run. G. Ue the formula m = run rie. H. Subtract x from both ide of the equation and divide every term by 6. I. Subtract from both ide of the equation and divide every term by.. You depoit $500 in a aving account that earn 6% imple interet per year. Auming you do not make any other depoit or withdrawal, how much interet will your account have earned after year? (7.RP.) A. $60 C. $600 B. $65 D. $650. What i the mean of the data in the box below? (7.NS.) 8.75,.,.5,.6, 5.97 F. 5 H. G. I. 5 Standardized Tet Practice 59

45 5. A football field i 0 yard wide and 0 yard long. Find the ditance between oppoite corner of the football field. Show your work and explain your reaoning. (8.G.7) 6. A compoite olid and it dimenion are hown below. 0 in. 8 in. in. 6 in. 6 in. What i the urface area, in quare inche, of the compoite olid? (7.G.6) 7. What i the olution to the equation hown below? (8.EE.7b) x = 6 A. C. B. D What i the value of x in the right triangle hown? (8.G.7) F. 6 cm H. cm x G. 8 cm I. 67 cm 7 cm 5 cm 9. Find the height of the tree in the diagram. (7.G.) A..5 ft C. 5 ft B..5 ft D. 0 ft 6 ft.5 ft 0 ft Not drawn to cale 550 Chapter Radical and the Pythagorean Theorem

46 0. Which expreion i equivalent to? (8.NS.) F. 8 6 H. 6 G. I. 6. The meaure of an angle i x degree. What i the meaure of it complement? (7.G.5) A. (90 x) C. (x 90) B. (80 x) D. (x 80). A pinner i divided into 6 congruent ection, a hown at the right. You pin the pinner. What i the probability that the arrow will land on a red ection? (7.SP.7a) F. 6 G. H. I.. An airplane flie 56 mile due north and then mile due eat. How many mile i the plane from it tarting point? (8.G.8). Which graph repreent the linear equation y = x? (8.EE.6) A. y x C. 5 y 5 x B. y x D. 5 y 5 x Standardized Tet Practice 55