8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

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1 8-1 he Pythagorean heorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number Square Positive Square Root Vocabulary Builder leg (noun) leg Related Word: hypotenuse Definition: In a right triangle, the sides that form the right angle are the legs. Main Idea: he legs of a right triangle are perpendicular. he hypotenuse is the side opposite the right angle. Use Your Vocabulary 2. Underline the correct word to complete the sentence. he hypotenuse is the longest / shortest side in a right triangle. Write for true or F for false. F 3. he hypotenuse of a right triangle can be any one of the three sides. 4. One leg of the triangle at the right has length 9 cm.. he hypotenuse of the triangle at the right has length 1 cm. leg hypotenuse leg 1 cm cm 9 cm Chapter 8 202

2 heorems 8-1 and 8-2 Pythagorean heorem and Its Converse Pythagorean heorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. If nabc is a right triangle, then a 2 1 b 2 c 2. Converse of the Pythagorean heorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. If a 2 1 b 2 c 2, then nabc is a right triangle. 6. Circle the equation that shows the correct relationship among the lengths of the legs and the hypotenuse of a right triangle Underline the correct words to complete each sentence. 7. A triangle with side lengths 3, 4, and is / is not a right triangle because is equal / not equal to A triangle with side lengths 4,, and 6 is / is not a right triangle because is equal / not equal to 6 2. A c b B a C Problem 1 Finding the Length of the Hypotenuse Got It? he legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? 9. Label the triangle at the right. 10. Use the justifications below to find the length of the hypotenuse. a 2 1 b 2 c 2 Pythagorean heorem c 2 Substitute for a and b c 2 Simplify c 2 Add. c ake the positive square root. 11. he length of the hypotenuse is 26.. One Pythagorean triple is,, and. If you multiply each number by 2, what numbers result? How do the numbers that result compare to the lengths of the sides of the triangle in Exercises 9 11? 10, 24, 26. Answers may vary. Sample: he numbers are the same 10 c 24 as the lengths of the sides of the triangle in Exercises Lesson 8-1

3 Problem 3 Finding Distance Got It? he size of a computer monitor is the length of its diagonal. You want to buy a 19-in. monitor that has a height of 11 in. What is the width of the monitor? Round to the nearest tenth of an inch. 19 in. 11 in.. Label the diagram of the computer monitor at the right. 14. he equation is solved below. Write a justification for each step. b in. a 2 1 b 2 c 2 Pythagorean heorem b Substitute. 1 1 b Simplify b b Subtract 1 from each side. Simplify. b "240 ake the positive square root. b < Use a calculator. 1. o the nearest tenth of an inch, the width of the monitor is 1. in. Problem 4 Identifying a Right riangle Got It? A triangle has side lengths 16, 48, and 0. Is the triangle a right triangle? Explain. 16. Circle the equation you will use to determine whether the triangle is a right triangle Simplify your equation from Exercise u Underline the correct words to complete the sentence. he equation is true / false, so the triangle is / is not a right triangle. A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a 2 1 b 2 c 2. If you multiply each number in a Pythagorean triple by the same whole number, the three numbers that result also form a Pythagorean triple. Chapter 8 204

4 heorems 8-3 and 8-4 Pythagorean Inequality heorems heorem 8-3 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. heorem 8-4 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. Use the figures at the right. Complete each sentence with acute or obtuse. 19. In nabc, c 2. a 2 1 b 2, so nabc is In nrs, s 2, r 2 1 t 2, so nrs is 9. obtuse acute A R t c b C S r s a B Lesson Check Do you UNDERSAND? Error Analysis A triangle has side lengths 16, 34, and 30. Your friend says it is not a right triangle. Look at your friend s work and describe the error. 21. Underline the length that your friend used as the longest side. Circle the length of the longest side of the triangle ? = 30 2? = Write the comparison that your friend should have used to determine whether the triangle is a right triangle Describe the error in your friend s work. Answers may vary. Sample: My friend used the wrong length for c in the comparison. he comparison should be Math Success Check off the vocabulary words that you understand. hypotenuse leg Pythagorean heorem Pythagorean triple Rate how well you can use the Pythagorean heorem and its converse. Need to review Now I get it! 20 Lesson 8-1

5 8-2 Special Right riangles Vocabulary Review 1. Circle the segment that is a diagonal of square ABCD. AB AC AD BC CD 2. Underline the correct word to complete the sentence. A diagonal is a line segment that joins two sides / vertices of a polygon. Vocabulary Builder D A C B complement (noun) KAHM pluh munt Other Word Form: complementary (adjective) Math Usage: When the measures of two angles have a sum of 90, each angle is a complement of the other. Nonexample: wo angles whose measures sum to 180 are supplementary. Use Your Vocabulary Complete each statement with the word complement or complementary. 3. If m/a 40 and m/b 0, the angles are 9. complementary 4. If m/a 30 and m/b 60, /B is the 9 of /A. complement. /P and /Q are 9 because the sum of their measures is 90. complementary Complete. 6. If /R has a measure of 3, then the complement of /R has a measure of. 7. If /X has a measure of 22, then the complement of /X has a measure of If /C has a measure of 6, then the complement of /C has a measure of Circle the complementary angles Chapter 8 206

6 heorem riangle heorem In a triangle, both legs are congruent and the length of the hypotenuse is "2 times the length of a leg. s 2 4 s Complete each statement for a triangle. 10. hypotenuse "2? leg 11. If leg 10, then hypotenuse "2? s Problem 1 Finding the Length of the Hypotenuse Got It? What is the length of the hypotenuse of a triangle with leg length!3?. Use the justifications to find the length of the hypotenuse. hypotenuse "2? leg riangle heorem "2? "3 Substitute. "2? "3 Commutative Property of Multiplication. "6 Simplify. Problem 2 Finding the Length of a Leg Got It? he length of the hypotenuse of a triangle is 10. What is the length of one leg?. Will the length of the leg be greater than or less than 10? Explain. Less than. Explanations may vary. Sample: he hypotenuse is the longest side. 14. Use the justifications to find the length of one leg. hypotenuse "2? leg 10 "2? leg Substitute riangle heorem 10 "2? leg Divide each side by "2. "2 "2 10 leg "2 Simplify. 10 "2 leg? "2 "2 Multiply by a form of 1 to rationalize the denominator. 10"2 leg 2 Simplify. leg "2 Divide by Lesson 8-2

7 Problem 3 Finding Distance Got It? You plan to build a path along one diagonal of a 100 ft-by- 100 ft square garden. o the nearest foot, how long will the path be? 1. Use the words path, height, and width to complete the diagram. 16. Write L for leg or H for hypotenuse to identify each part of the right triangle in the diagram. H path L height L width 17. Substitute for hypotenuse and leg. Let h the length of the hypotenuse. hypotenuse "2? leg h "2? Solve the equation. Use a calculator to find the length of the path. h!2? 100 h N height width path 19. o the nearest foot, the length of the path will be 141 feet. heorem riangle heorem In a triangle, the length of the hypotenuse is twice the length of the shorter leg. he length of the longer leg is "3 times the length of the shorter leg. Complete each statement for a triangle. 20. hypotenuse 2? shorter leg 21. longer leg "3? shorter leg Problem 4 hink f is the length of the hypotenuse. I can write an equation relating the hypotenuse and the 3 shorter leg of the triangle. 3 Now I can solve for f. Using the Length of One Side Got It? What is the value of f in simplest radical form? 22. Complete the reasoning model below. hypotenuse f f Write 2 2 shorter leg œ s s 30 f s V3 Chapter 8 208

8 Problem Applying the riangle heorem Got It? Jewelry Making An artisan makes pendants in the shape of equilateral triangles. Suppose the sides of a pendant are 18 mm long. What is the height of the pendant to the nearest tenth of a millimeter? 18 mm 18 mm 23. Circle the formula you can use to find the height of the pendant. hypotenuse 2? shorter leg 24. Find the height of the pendant. longer leg!3? shorter leg 18 mm longer leg "3? shorter leg "3? 9 N o the nearest tenth of a millimeter, the height of the pendant is 1.6 mm. Lesson Check Do you UNDERSAND? Reasoning A test question asks you to find two side lengths of a triangle. You know that the length of one leg is 6, but you forgot the special formula for triangles. Explain how you can still determine the other side lengths. What are the other side lengths? 26. Underline the correct word(s) to complete the sentence. In a triangle, the lengths of the legs are different / the same. 27. Use the Pythagorean heorem to find the length of the longest side. 28. he other two side lengths are 6 and 6"2. Math Success Check off the vocabulary words that you understand. leg hypotenuse right triangle Pythagorean heorem Rate how well you can use the properties of special right triangles. Need to review longest side: c c c 2 72 c "72 6" Now I get it! 209 Lesson 8-2

9 8-3 rigonometry Vocabulary Review he Venn diagram at the right shows the relationship between similar and congruent figures. Write for true or F for false. F 1. All similar figures are congruent figures. 2. All congruent figures are similar figures. Similar Figures Congruent Figures 3. Some similar figures are congruent figures. 4. Circle the postulate or theorem you can use to verify that the triangles at the right are similar. AA, Postulate SAS, heorem SSS, heorem Vocabulary Builder ratio (noun) RAY shee oh Related Words: rate, rational Definition: A ratio is the comparison of two quantities by division. Example: If there are 6 triangles and squares, the ratio of triangles to squares is 6 and the ratio of square to triangles is 6. Use Your Vocabulary Use the triangle at the right for Exercises 7.. Circle the ratio of the length of the longer leg to the length of the shorter leg. 6. Circle the ratio of the length of the shorter leg to the length of the hypotenuse. 7. Circle the ratio of the length of the longer leg to the length of the hypotenuse. Chapter 8 210

10 Key Concept he rigonometric Ratios sine of /A cosine of /A tangent of /A length of leg opposite/a a length of hypotenuse c length of leg adjacent to/a b length of hypotenuse c length of leg opposite/a length of leg adjacent to/a a b A c b B a C Draw a line from each trigonometric ratio in Column A to its corresponding ratio in Column B. Column A 8. sin B 9. cos B 10. tan B Column B 11. Reasoning Suppose nabc is a right isosceles triangle. What would the tangent of /B equal? Explain. Explanations may vary. Sample: a c b a b c 1. he legs would be congruent, so b a would equal 1. Problem 1 Writing rigonometric Ratios Got It? What are the sine, cosine, and tangent ratios for lg?. Circle the measure of the leg opposite /G Circle the measure of the hypotenuse Circle the measure of the leg adjacent to /G Write each trigonometric ratio. sin G cos G opposite hypotenuse adjacent hypotenuse tan G opposite adjacent G 8 R 211 Lesson 8-3

11 Problem 2 Using a rigonometric Ratio to Find Distance Got It? Find the value of w to the nearest tenth. Below is one student s solution w cos 4 w 17 cos 4 (17) w w 10 w 16. Circle the trigonometric ratio that uses sides w and 17. sin 48 cos 48 tan What error did the student make? Answers may vary. Sample: he student wrote cos 4 w 17 rather than sin 4 w Find the value of w correctly. sin 4 w 17 sin 4 (17) w N w.8 N w 19. he value of w to the nearest tenth is.8. Problem 3 Using Inverses Got It? Use the figure below. What is mly to the nearest degree? P Y 20. Circle the lengths that you know. hypotenuse side adjacent to /Y side opposite /Y 21. Cross out the ratios that you will NO use to find m/y. sine cosine tangent 22. Underline the correct word to complete the statement. If you know the sine, cosine, or tangent ratio of an angle, you can use the inverse / ratio to find the measure of the angle. Chapter 8 2

12 23. Follow the steps to find m/y. 1 Write the ratio. 100 tan Y 41 2 Use the inverse. 100 Y tan ( 1 41 ) 3 Use a calculator. Y o the nearest degree, m/y < 68. Lesson Check Do you UNDERSAND? Error Analysis A student states that sin A S sin X because the lengths of the sides of kabc are greater than the lengths of the sides of kxyz. What is the student s error? Explain. Y B Underline the correct word(s) to complete each sentence. 2. nabc and nxyz are / are not similar. Z 3 X C 3 A 26. /A and /X are / are not congruent, so sin 38 is / is not equal to sin What is the student s error? Explain. Answers may vary. Sample: he student did not look at the measures of la and lx. Congruent angles have equal sine ratios. Math Success Check off the vocabulary words that you understand. trigonometric ratios sine cosine tangent Rate how well you can use trigonometric ratios. Need to review Now I get it! 2 Lesson 8-3

13 8-4 Angles of Elevation and Depression Vocabulary Review Underline the correct word(s) or number to complete each sentence. 1. he measure of a right angle is greater / less than the measure of an acute angle and greater / less than the measure of an obtuse angle. 2. A right angle has a measure of 4 / 90 / Lines that intersect to form four right angles are parallel / perpendicular lines. 4. Circle the right angle(s) in the figure. /ACB /ADB /BAC A /BAD /CBA /DBA Vocabulary Builder D B C elevation (noun) el uh VAY shun Related Word: depression Definition: he elevation of an object is its height above a given level, such as eye level or sea level. Math Usage: Angles of elevation and depression are acute angles of right triangles formed by a horizontal distance and a vertical height. Use Your Vocabulary Complete each statement with the correct word from the list below. Use each word only once. elevate elevated elevation. John 9 his feet on a footstool. 6. he 9 of Mt McKinley is 20,320 ft. 7. You 9 an object by raising it to a higher position. elevated elevation elevate Chapter 8 214

14 Problem 1 Identifying Angles of Elevation and Depression Got It? What is a description of l2 as it relates to the situation shown? Write for true or F for false. 8. /2 is above the horizontal line. F 9. /2 is the angle of elevation from the person in the hot-air balloon to the bird. 10. /2 is the angle of depression from the person in the hot-air balloon to the bird. F 11. /2 is the angle of elevation from the top of the mountain to the person in the hot-air balloon.. Describe /2 as it relates to the situation shown. Answers may vary. Sample: l2 is the angle of elevation from the person in the hot-air balloon to the bird. Problem 2 Using the Angle of Elevation Got It? You sight a rock climber on a cliff at a 32 angle of elevation. Your eye level is 6 ft above the ground and you are 1000 feet from the base of the cliff. What is the approximate height of the rock climber from the ground?. Use the information in the problem to complete the problem-solving model below. Eye level ft Know Need Plan Angle of elevation Height of climber from Find the length of the is the ground leg opposite 328 by using tan Distance to the cliff is 1000 ft. hen add 6 ft. Eye level is 6 above the ground. ft Climber 21 Lesson 8-4

15 14. Explain why you use tan 328 and not sin 328 or cos 328. Answers may vary. Sample: he sine ratio involves two unknowns. he cosine ratio involves the hypotenuse and 1000, but I do not want to know the hypotenuse. he ratio that uses the unknown height and 1000 is the tangent ratio. 1. he problem is solved below. Use one of the reasons from the list at the right to justify each step. tan 328 d 1000 Write the equation. Solve for d. Use a calculator. Write the equation. (tan 328) 1000 d Solve for d. d < Use a calculator. 16. he height from your eye level to the climber is about 62 ft. 17. he height of the rock climber from the ground is about 631 ft. Problem 3 Using the Angle of Depression Got It? An airplane pilot sights a life raft at a 26 angle of depression. he airplane s altitude is 3 km. What is the airplane s horizontal distance d from the raft? 18. Label the diagram below. altitude 3 km Not to scale 26º Angle of elevation 26º d Angle of depression horizontal distance Raft 19. Circle the equation you could use to find the horizontal distance d. sin cos d d 20. Solve your equation from Exercise 19. tan d d 3 tan 268 d tan d 21. o the nearest tenth, the airplane s horizontal distance from the raft is 6.2 km. Chapter 8 216

16 Lesson Check Do you UNDERSAND? Vocabulary How is an angle of elevation formed? Underline the correct word(s) to complete each sentence. 22. he angle of elevation is formed above / below a horizontal line. 23. he angle of depression is formed above / below a horizontal line. 24. he measure of an angle of elevation is equal to / greater than / less than the measure of the angle of depression. Lesson Check Do you UNDERSAND? Error Analysis A homework question says that the angle of depression from the bottom of a house window to a ball on the ground is 20. At the right is your friend s sketch of the situation. Describe your friend s error. 2. Is the angle that your friend identified as the angle of depression formed by the horizontal and the line of sight? Yes / No 26. Is the correct angle of depression adjacent to or opposite the angle identified by your friend? 27. Describe your friend s error. Math Success Check off the vocabulary words that you understand. angle of elevation angle of depression trigonometric ratios Rate how well you can use angles of elevation and depression. Need to review Now I get it! 20 adjacent to / opposite Answers may vary. Sample: My friend identified the wrong angle. he correct angle of depression is below the horizontal line. 217 Lesson 8-4

17 8- Vectors Vocabulary Review 1. Circle the drawing that shows only segment AB. A B A B A B Use the number line below to find the length of each segment. A B C D AB 1 3. AC 4 4. BC 3. BD 6. Explain how a line segment is different from a line. Explanations may vary. Sample: A line segment has endpoints. A line does not have endpoints and extends without end. Vocabulary Builder vector (noun) VEK tur Related Words: magnitude, direction Definition: A vector is any quantity with magnitude (size) and direction. Main Idea: You can use vectors to model motion and direction. Example: A car s speed and direction together represent a vector. Use Your Vocabulary Write for true or F for false. F 7. A vector has an initial point and a terminal point. 8. he terminal point of the vector at the right is point O. 9. In symbols, vector OB is written as OB W. y P O x Vector OP, or OP y O x B Chapter 8 218

18 Problem 1 Describing a Vector Got It? What is the vector at the right as an ordered pair? Round the coordinates to the nearest tenth. 10. Label the diagram with the lengths x and y. y x 3 10 y O x 11. Circle the part of the triangle that has a length of 3. leg opposite leg adjacent to hypotenuse 10 -angle 10 -angle. Circle the part of the triangle that has length x. leg opposite leg adjacent to hypotenuse 10 -angle 10 -angle. Circle the part of the triangle that has length y. leg opposite leg adjacent to hypotenuse 10 -angle 10 -angle 14. Use the justifications below to find the values of x and y. cos 10 x Write the ratios. sin 10 y 1 3 3? cos 10 x Solve for x and y. 3? sin 10 y < x Use a calculator < y < x Round to the nearest tenth. 4.2 < y 1. Decide whether each coordinate is positive or negative. x-coordinate: 9 y-coordinate: he coordinates of the vector are k , 24 l. Problem 2 negative negative Describing a Vector Direction Got It? What is the direction of the vector at the right? 17. Is the angle above (north) or below (south) the above / below west-east line? 18. Is the angle to the left (west) or to the right (east) of left / right the north-south line? 19. Circle the direction of the vector. 60 south 60 north 60 south 60 north of east of east of west of west W 60 N S E 219 Lesson 8-

19 Problem 3 Finding the Magnitude and Direction of a Vector Got It? An airplane lands 246 mi east and 76 mi north from where it took off. What are the approximate magnitude and direction of its flight vector? N 20. Label the diagram with the lengths 246 and he vector k 246, 76 l describes the result of the trip. 22. Complete the reasoning model below. W S d 76 x E 246 hink he magnitude is the distance from the initial point to the terminal point. I can use the Distance Formula to find the distance between (0, 0) and (246, 76) Write d (246 0) 2 (76 0) 2 60, , he vector is x north of east. I can use the tangent ratio to find this angle formed by the vector. hen I can use a calculator to find the inverse tangent. 76 tan x 246 x tan x he magnitude is about 27 mi and the direction is about 178 north of east. Property Adding Vectors For a W kx 1, y 1 l and c W kx 2, y 2 l, a W 1 c W kx 1 1 x 2, y 1 1 y 2 l Problem 4 Adding Vectors Got It? What is the resultant of k2, 3l and k24, 22 l as an ordered pair? 24. he sum is found below. Use one of the reasons in the list to justify each step. e W a W 1 c W W e k2, 3l 1 k24, 22l W e k2 1 (24), 3 1 (22)l Write the sum. Substitute. Write the sum. Substitute. Simplify. Add the coordinates. Add the coordinates. e W k22, 1l Simplify. Chapter 8 220

20 Problem Applying Vectors Got It? Reasoning he speed of a powerboat in still water is 3 mi/h. he river flows directly south at 8 mi/h. At what angle should the powerboat head up river in order to travel directly west? 2. Label the sides of the triangle in the diagram. 26. Use trigonometry to find x. sin x 8 3 x sin 21 Q 8 3 R x N mi/h 8 mi/h x W N E S boat 27. he angle at which the powerboat should head up river is about.2. Lesson Check Do you UNDERSAND? Error Analysis Your friend says that the magnitude of vector k10, 7l is greater than that of vector k210, 27l because the coordinates of k10, 7l are positive and the coordinates of k210, 27l are negative. Explain why your friend s statement is incorrect. 28. Complete to find the magnitude of each vector. d 1 "(10 2 0) 2 1 (7 2 0) 2 d 2 "( ) 2 1 (27 2 0) 2 2 Å Å (210) 2 1 ( 27 ) 2 Math Success Check off the vocabulary words that you understand. vector magnitude initial point terminal point resultant Rate how well you can use and describe vectors. Need to review Å Å Å 149 Å Explain why your friend s statement is incorrect. Explanations may vary. Sample: When using the Distance Formula to find magnitude, you square the coordinates (210) 2 1 ( 27) 2, so the magnitudes are equal Now I get it! 221 Lesson 8-