PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side

VOCABULARY Copy and complete: A set of three positive integers a, b, and c that satisfy the equation c2 5 a2 1 b2 is called a ? .

WRITING Describe the information you need to have in order to use the Pythagorean Theorem to find the length of a side of a triangle.

ALGEBRA Find the length of the hypotenuse of the right triangle. 120 5

ALGEBRA Find the length of the hypotenuse of the right triangle. x 56 33

ALGEBRA Find the length of the hypotenuse of the right triangle. 42 40

ERROR ANALYSIS Describe and correct the error in using the Pythagorean Theorem. 2 1 b2 5 c2 102 1 262 5 242 26

ERROR ANALYSIS Describe and correct the error in using the Pythagorean Theorem. x2 5 72 1 242 x2 5 (7 1 24)2 x2 5 312 x 5 31 x 24 7

FINDING A LENGTH Find the unknown leg length x. 16.7 ft x 8.9 ft

FINDING A LENGTH Find the unknown leg length x. 9.8 in. 13.4 in. x

FINDING A LENGTH Find the unknown leg length x. 5.7 ft 4.9 ft x

FINDING THE AREA Find the area of the isosceles triangle. 7 m 17 m 16 m

FINDING THE AREA Find the area of the isosceles triangle. 20 ft 20 ft 32 ft

FINDING THE AREA Find the area of the isosceles triangle. 10 cm 10 cm 12 cm

FINDING SIDE LENGTHS Find the unknown side length of the right triangle using the Pythagorean Theorem or a Pythagorean triple. 72 21

FINDING SIDE LENGTHS Find the unknown side length of the right triangle using the Pythagorean Theorem or a Pythagorean triple. 50 30

FINDING SIDE LENGTHS Find the unknown side length of the right triangle using the Pythagorean Theorem or a Pythagorean triple. 60 68

MULTIPLE CHOICE What is the length of the hypotenuse of a right triangle with leg lengths of 8 inches and 15 inches? A 13 inches B 17 inches C 21 inches D 25 inches

PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse. 24 and 51

PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse. . 20 and 25

PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse. 28 and 96

PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse. . 20 and 48

PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse. 75 and 85

PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse. 72 and 75

FINDING SIDE LENGTHS Find the unknown side length x. Write your answer in simplest radical form. 6 3 6

FINDING SIDE LENGTHS Find the unknown side length x. Write your answer in simplest radical form. 11 x

FINDING SIDE LENGTHS Find the unknown side length x. Write your answer in simplest radical form. 3 7 5

MULTIPLE CHOICE What is the area of a right triangle with a leg length of 15 feet and a hypotenuse length of 39 feet? A 270 ft2 B 292.5 ft2 C 540 ft2 D 585 ft2

ALGEBRA Solve for x if the lengths of the two legs of a right triangle are 2x and 2x 1 4, and the length of the hypotenuse is 4x 2 4.

CHALLENGE In Exercises 29 and 30, solve for x. 39 9 10 3

CHALLENGE In Exercises 29 and 30, solve for x. 3 15 14

BASEBALL DIAMOND In baseball, the distance of the paths between each pair of consecutive bases is 90 feet and the paths form right angles. How far does the ball need to travel if it is thrown from home plate directly to second base?

APPLE BALLOON You tie an apple balloon to a stake in the ground. The rope is 10 feet long. As the wind picks up, you observe that the balloon is now 6 feet away from the stake. How far above the ground is the balloon now?

SHORT RESPONSE Three side lengths of a right triangle are 25, 65, and 60. Explain how you know which side is the hypotenuse.

MULTI-STEP PROBLEM In your town, there is a field that is in the shape of a right triangle with the dimensions shown. a. Find the perimeter of the field. b. You are going to plant dogwood seedlings about every ten feet around the fields edge. How many trees do you need? c. If each dogwood seedling sells for $12, how much will the trees cost? 35 ft x ft 80 ft

MULTIPLE REPRESENTATIONS As you are gathering leaves for a science project, you look back at your campsite and see that the campfire is not completely out. You want to get water from a nearby river to put out the flames with the bucket you are using to collect leaves. Use the diagram and the steps below to determine the shortest distance you must travel. a. Making a Table Make a table with columns labeled BC, AC, CE, and AC 1 CE. Enter values of BC from 10 to 120 in increments of 10. b. Calculating Values Calculate AC, CE, and AC 1 CE for each value of BC, and record the results in the table. Then, use your table of values to determine the shortest distance you must travel. c. Drawing a Picture Draw an accurate picture to scale of the shortest distance. A C D E 120 ft 30 ft

SHORT RESPONSE Justify the Distance Formula using the Pythagorean Theorem.

PROVING THEOREM 4.5 Find the Hypotenuse-Leg (HL) Congruence Theorem on page 241. Assign variables for the side lengths in the diagram. Use your variables to write GIVEN and PROVE statements. Use the Pythagorean Theorem and congruent triangles to prove Theorem 4.5.

CHALLENGE Trees grown for sale at nurseries should stand at least five feet from one another while growing. If the trees are grown in parallel rows, what is the smallest allowable distance between rows?

Evaluate the expression. (p. 874) 1} 7 2 2

Evaluate the expression. (p. 874) 14} 3 2 2

Evaluate the expression. (p. 874) 126} 81 2 2

Evaluate the expression. (p. 874) 128} 2 2 2

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. (p. 328) 3 feet, 6 feet

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. (p. 328) 5 inches, 11 inches

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. (p. 328) 14 meters, 21 meters

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. (p. 328) 12 inches, 27 inches

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. (p. 328) 18 yards, 18 yards

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. (p. 328) 27 meters, 39 meters

Determine whether the two triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A. (p. 388) 5 8 10 A B

Determine whether the two triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A. (p. 388) 6 10 12 A B

VOCABULARY What is the longest side of a right triangle called?

WRITING Explain how the side lengths of a triangle can be used to classify it as acute, right, or obtuse.

VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle. 65 97

VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle. 21.2 11.4

VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle. 2 6 3 5

VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle. 14 10 4

VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle. 1 2

VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle. 80 39 8

VERIFYING RIGHT TRIANGLES Tell whether the given side lengths of a triangle can represent a right triangle. 9, 12, and 15

VERIFYING RIGHT TRIANGLES Tell whether the given side lengths of a triangle can represent a right triangle. 9, 10, and 15

VERIFYING RIGHT TRIANGLES Tell whether the given side lengths of a triangle can represent a right triangle. 36, 48, and 60

VERIFYING RIGHT TRIANGLES Tell whether the given side lengths of a triangle can represent a right triangle. 6, 10, and 2 } 34

VERIFYING RIGHT TRIANGLES Tell whether the given side lengths of a triangle can represent a right triangle. 7, 14, and 7 } 5

VERIFYING RIGHT TRIANGLES Tell whether the given side lengths of a triangle can represent a right triangle. 10, 12, and 20

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 10, 11, and 14

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 10, 15, and 5 } 13

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 24, 30, and 6 } 43

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 5, 6, and 7

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 12, 16, and 20

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 8, 10, and 12

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 15, 20, and 36

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 6, 8, and 10

CLASSIFYING TRIANGLES In Exercises 1523, decide if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? 8.2, 4.1, and 12.2

MULTIPLE CHOICE Which side lengths do not form a right triangle? A 5, 12, 13 B 10, 24, 28 C 15, 36, 39 D 50, 120, 130

MULTIPLE CHOICE What type of triangle has side lengths of 4, 7, and 9? A Acute scalene B Right scalene C Obtuse scalene D None of the above

ERROR ANALYSIS A student tells you that if you double all the sides of a right triangle, the new triangle is obtuse. Explain why this statement is incorrect.

GRAPHING TRIANGLES Graph points A, B, and C. Connect the points to form nABC. Decide whether nABC is acute, right, or obtuse. A(22, 4), B(6, 0), C(25, 22)

GRAPHING TRIANGLES Graph points A, B, and C. Connect the points to form nABC. Decide whether nABC is acute, right, or obtuse. A(0, 2), B(5, 1), C(1, 21)

ALGEBRA Tell whether a triangle with side lengths 5x, 12x, and 13x (where x > 0) is acute, right, or obtuse.

USING DIAGRAMS In Exercises 30 and 31, copy and complete the statement with <, >, or 5, if possible. If it is not possible, explain why. 18 4 10 2 96 F B m A ? m D

USING DIAGRAMS In Exercises 30 and 31, copy and complete the statement with <, >, or 5, if possible. If it is not possible, explain why. 18 4 10 2 96 F B m B 1 m C ? m E 1 m F

OPEN-ENDED MATH The side lengths of a triangle are 6, 8, and x (where x > 0). What are the values of x that make the triangle a right triangle? an acute triangle? an obtuse triangle?

ALGEBRA The sides of a triangle have lengths x, x 1 4, and 20. If the length of the longest side is 20, what values of x make the triangle acute?

CHALLENGE The sides of a triangle have lengths 4x 1 6, 2x 1 1, and 6x 2 1. If the length of the longest side is 6x 2 1, what values of x make the triangle obtuse?

PAINTING You are making a canvas frame for a painting using stretcher bars. The rectangular painting will be 10 inches long and 8 inches wide. Using a ruler, how can you be certain that the corners of the frame are 908?

WALKING You walk 749 feet due east to the gym from your home. From the gym you walk 800 feet southwest to the library. Finally, you walk 305 feet from the library back home. Do you live directly north of the library? Explain.

MULTI-STEP PROBLEM Use the diagram shown. a. Find BC. b. Use the Converse of the Pythagorean Theorem to show that nABC is a right triangle. c. Draw and label a similar diagram where nDBC remains a right triangle, but nABC is not. B 13 12 3 4 C A

SHORT RESPONSE You are setting up a volleyball net. To stabilize the pole, you tie one end of a rope to the pole 7 feet from the ground. You tie the other end of the rope to a stake that is 4 feet from the pole. The rope between the pole and stake is about 8 feet 4 inches long. Is the pole perpendicular to the ground? Explain. If it is not, how can you fix it? 8 ft 4 in 4 ft 7 ft

EXTENDED RESPONSE You are considering buying a used car. You would like to know whether the frame is sound. A sound frame of the car should be rectangular, so it has four right angles. You plan to measure the shadow of the car on the ground as the sun shines directly on the car. a. You make a triangle with three tape measures on one corner. It has side lengths 12 inches, 16 inches, and 20 inches. Is this a right triangle? Explain. b. You make a triangle on a second corner with side lengths 9 inches, 12 inches, and 18 inches. Is this a right triangle? Explain. c. The car owner says the car was never in an accident. Do you believe this claim? Explain.

PROVING THEOREM 7.3 Copy and complete the proof of Theorem 7.3. GIVEN c In nABC, c2 < a2 1 b2 where c is the length of the longest side. PROVE c nABC is an acute triangle. Plan for Proof Draw right nPQR with side lengths a, b, and x, where R is a right angle and x is the length of the longest side. Compare lengths c and x. STATEMENTS REASONS 1. In nABC, c2 < a2 1 b2 where c is the length of the longest side. In nPQR, R is a right angle. 2. a2 1 b2 5 x2 3. c2 < x2 4. c < x 5. m R 5 908 6. m C < m ? 7. m C < 908 8. C is an acute angle. 9. nABC is an acute triangle. 1. ? 2. ? 3. ? 4. A property of square roots 5. ? 6. Converse of the Hinge Theorem 7. ? 8. ? 9. ? c b C A B x b R P P

PROVING THEOREM 7.4 Prove Theorem 7.4. Include a diagram and GIVEN and PROVE statements. (Hint: Look back at Exercise 40.)

PROVING THEOREM 7.2 Prove the Converse of the Pythagorean Theorem. GIVEN c In nLMN,}LM is the longest side, and c2 5 a2 1 b2 . PROVE c nLMN is a right triangle. Plan for Proof Draw right nPQR with side lengths a, b, and x. Compare lengths c and x. b x a R P b P c

SHORT RESPONSE Explain why D must be a right angle. 0 6 9 C D B E A

COORDINATE PLANE Use graph paper. a. Graph nABC with A(27, 2), B(0, 1) and C(24, 4). b. Use the slopes of the sides of nABC to determine whether it is a right triangle. Explain. c. Use the lengths of the sides of nABC to determine whether it is a right triangle. Explain. d. Did you get the same answer in parts (b) and (c)? If not, explain why.

CHALLENGE Find the values of x and y. 3 3 5 5 4

In Exercises 4648, copy the triangle and draw one of its altitudes. (p. 319)

In Exercises 4648, copy the triangle and draw one of its altitudes. (p. 319)

In Exercises 4648, copy the triangle and draw one of its altitudes. (p. 319)

Copy and complete the statement. (p. 364) If 10 }x 5 7 }y , then 10 }7 5 ? }?

Copy and complete the statement. (p. 364) f x }15 5 y }2 , then x }y 5 ? }?

Copy and complete the statement. (p. 364) If x }8 5 y }9 , then x } 1 8 8 5 ? }?

The perimeter of a rectangle is 135 feet. The ratio of the length to the width is 8 : 1. Find the length and the width. (p. 372)

VOCABULARY Copy and complete: Two triangles are ? if their corresponding angles are congruent and their corresponding side lengths are proportional.

WRITING In your own words, explain geometric mean.

IDENTIFYING SIMILAR TRIANGLES Identify the three similar right triangles in the given diagram. H G F

IDENTIFYING SIMILAR TRIANGLES Identify the three similar right triangles in the given diagram. L N K M

FINDING ALTITUDES Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 107.5 ft 76 ft 76 ft

FINDING ALTITUDES Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 26.6 ft 23 ft 12.8 ft

FINDING ALTITUDES Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 13.2 ft 10 ft 8.8 ft x

COMPLETING PROPORTIONS Write a similarity statement for the three similar triangles in the diagram. Then complete the proportion. XW ? 5 } ZW YW Z Y X W

COMPLETING PROPORTIONS Write a similarity statement for the three similar triangles in the diagram. Then complete the proportion. ? SQ 5 SQ }TQ S R P

COMPLETING PROPORTIONS Write a similarity statement for the three similar triangles in the diagram. Then complete the proportion. EF EG 5 } EG ? H F G E

ERROR ANALYSIS Describe and correct the error in writing a proportion for the given diagram. w }z 5} z w 1 v z w v x

ERROR ANALYSIS Describe and correct the error in writing a proportion for the given diagram. e }d 5 d }f f h d g

FINDING LENGTHS Find the value of the variable. Round decimal answers to the nearest tenth. x 5 4

FINDING LENGTHS Find the value of the variable. Round decimal answers to the nearest tenth. 12 y 18

FINDING LENGTHS Find the value of the variable. Round decimal answers to the nearest tenth. 6 27 z

FINDING LENGTHS Find the value of the variable. Round decimal answers to the nearest tenth. 4 9 x

FINDING LENGTHS Find the value of the variable. Round decimal answers to the nearest tenth. 5 8 y

FINDING LENGTHS Find the value of the variable. Round decimal answers to the nearest tenth. x 2 8

MULTIPLE CHOICE Use the diagram at the right. Decide which proportion is false. A } DB DC 5 } DA DB B } CA AB 5 } AB AD C } CA BA 5 } BA CA D } DC BC 5 } BC CA B C A

MULTIPLE CHOICE In the diagram in Exercise 19 above, AC 5 36 and BC 5 18. Find AD. If necessary, round to the nearest tenth. A 9 B 15.6 C 27 D 31.2

ALGEBRA Find the value(s) of the variable(s). a 1 5 18 12

ALGEBRA Find the value(s) of the variable(s). 8 b 1 3 6

ALGEBRA Find the value(s) of the variable(s). y 16 12

USING THEOREMS Tell whether the triangle is a right triangle. If so, find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 2 89 16 1

USING THEOREMS Tell whether the triangle is a right triangle. If so, find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 4 13 12 8

USING THEOREMS Tell whether the triangle is a right triangle. If so, find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 14 18 4 33

FINDING LENGTHS Use the Geometric Mean Theorems to find AC and BD. C B A D 20

CHALLENGE Draw a right isosceles triangle and label the two leg lengths x. Then draw the altitude to the hypotenuse and label its length y. Now draw the three similar triangles and label any side length that is equal to either x or y. What can you conclude about the relationship between the two smaller triangles? Explain

DOGHOUSE The peak of the doghouse shown forms a right angle. Use the given dimensions to find the height of the roof. 1.5 ft x 1.5

MONUMENT You want to determine the height of a monument at a local park. You use a cardboard square to line up the top and bottom of the monument. Mary measures the vertical distance from the ground to your eye and the distance from you to the monument. Approximate the height of the monument (as shown at the left below).

SHORT RESPONSE Paul is standing on the other side of the monument in Exercise 30 (as shown at the right above). He has a piece of rope staked at the base of the monument. He extends the rope to the cardboard square he is holding lined up to the top and bottom of the monument. Use the information in the diagram above to approximate the height of the monument. Do you get the same answer as in Exercise 30? Explain.

PROVING THEOREM 7.1 Use the diagram of nABC. Copy and complete the proof of the Pythagorean Theorem. GIVEN c In nABC, BCA is a right angle. PROVE c c2 5 a2 1 b2 STATEMENTS REASONS 1. Draw nABC. BCA is a right angle. 2. Draw a perpendicular from C to }AB . 3. c }a 5 a }e and c }b 5 b }f 4. ce 5 a2 and cf 5 b2 5. ce 1 b2 5 ? 1 b2 6. ce 1 cf 5 a2 1 b2 7. c(e 1 f) 5 a2 1 b2 8. e 1 f 5 ? 9. c p c 5 a2 1 b2 10. c2 5 a2 1 b2 1. ? 2. Perpendicular Postulate 3. ? 4. ? 5. Addition Property of Equality 6. ? 7. ? 8. Segment Addition Postulate 9. ? 10. Simplify. c b f e B D C A

MULTI-STEP PROBLEM Use the diagram. a. Name all the altitudes in nEGF. Explain. b. Find FH. c. Find the area of the triangle. F E 5 7

EXTENDED RESPONSE Use the diagram. a. Sketch the three similar triangles in the diagram. Label the vertices. Explain how you know which vertices correspond. b. Write similarity statements for the three triangles. c. Which segments length is the geometric mean of RT and RQ? Explain your reasoning. T R S P

PROVING THEOREMS In Exercises 3537, use the diagram and GIVEN statements below. GIVEN c nABC is a right triangle. Altitude }CD is drawn to hypotenuse }AB D B C A Prove Theorem 7.5 by using the Plan for Proof on page 449.

PROVING THEOREMS In Exercises 3537, use the diagram and GIVEN statements below. GIVEN c nABC is a right triangle. Altitude }CD is drawn to hypotenuse }AB D B C A Prove Theorem 7.6 by showing } BD CD 5 } CD AD

PROVING THEOREMS In Exercises 3537, use the diagram and GIVEN statements below. GIVEN c nABC is a right triangle. Altitude }CD is drawn to hypotenuse }AB D B C A Prove Theorem 7.7 by showing AB }}CB 5 CB }}DB and } AB AC 5 } AC AD .

CHALLENGE The harmonic mean of a and b is 2ab }}} a 1 b . The Greek mathematician Pythagoras found that three equally taut strings on stringed instruments will sound harmonious if the length of the middle string is equal to the harmonic mean of the lengths of the shortest and longest string. a. Find the harmonic mean of 10 and 15. b. Find the harmonic mean of 6 and 14. c. Will equally taut strings whose lengths have the ratio 4 : 6 : 12 sound harmonious? Explain your reasoning

Simplify the expression. (p. 874) } 27 p } 2

Simplify the expression. (p. 874) } 8 p } 10

Simplify the expression. (p. 874) } 12 p } 7

Simplify the expression. (p. 874) } 18 p } 12

Simplify the expression. (p. 874) 5 } 7

Simplify the expression. (p. 874) 8 } 11

Simplify the expression. (p. 874) 15 } 27

Simplify the expression. (p. 874) 12 } 24

Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. (p. 171) Line 1: (2, 4), (4, 2) Line 2: (3, 5), (21, 1)

Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. (p. 171) Line 1: (0, 2), (21, 21) Line 2: (3, 1), (1, 25)

Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. (p. 171) Line 1: (1, 7), (4, 7) Line 2: (5, 2), (7, 4)

VOCABULARY Copy and complete: A triangle with two congruent sides and a right angle is called ? .

WRITING Explain why the acute angles in an isosceles right triangle always measure 458.

458-458-908 TRIANGLES Find the value of x. Write your answer in simplest radical form. 7 4

458-458-908 TRIANGLES Find the value of x. Write your answer in simplest radical form. 5 2 5 2

458-458-908 TRIANGLES Find the value of x. Write your answer in simplest radical form. x x 3

MULTIPLE CHOICE Find the length of }AC . A 7 } 2 in. B 2 } 7 in. C 7} }2 2 in. D } 14 in. A 7 in. B C 458

ISOSCELES RIGHT TRIANGLE The square tile shown has painted corners in the shape of congruent 458-458-908 triangles. What is the value of x? What is the side length of the tile?

308-608-908 TRIANGLES Find the value of each variable. Write your answers in simplest radical form. 9 308 x

308-608-908 TRIANGLES Find the value of each variable. Write your answers in simplest radical form. x 608 3 3

308-608-908 TRIANGLES Find the value of each variable. Write your answers in simplest radical form. y x 308 12 3

SPECIAL RIGHT TRIANGLES Copy and complete the table. a b 458 4 a 7 ? ? ? } 5 b ? 11 ? ? ? C ? ? 10 6 } 2 ?

SPECIAL RIGHT TRIANGLES Copy and complete the table. e f d 308 6 d 5???? e ? ? 8 } 3 ? 12 f ? 14 ? 18 } 3 ?

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. y x 608 1

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 6 m

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 308 24 q

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 8 608 18 r

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 458 3 4 t u

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 458 608 e

MULTIPLE CHOICE Which side lengths do not represent a 308-608-908 triangle? A 1 }2 , } } 3 2 , 1 B } 2 , } 6 , 2} 2 C 5 }2 , 5} }3 2 , 10 D 3, 3} 3 , 6

ERROR ANALYSIS Describe and correct the error in finding the length of the hypotenuse. 7 30

ERROR ANALYSIS Describe and correct the error in finding the length of the hypotenuse. 5 2 5 5 45

WRITING Abigail solved Example 5 on page 459 in a different way. Instead of dividing each side by } 3 , she multiplied each side by } 3 . Does her method work? Explain why or why not.

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 10 f g 3

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. x 1508 4 2

ALGEBRA Find the value of each variable. Write your answers in simplest radical form. x 608 3

CHALLENGE nABC is a 308-608-908 triangle. Find the coordinates of A. y 1 1 C(23, 21) B(3, 21)

KAYAK RAMP A ramp is used to launch a kayak. What is the height of an 11 foot ramp when its angle is 308 as shown? 308 11 ft h

DRAWBRIDGE Each half of the drawbridge is about 284 feet long, as shown. How high does a seagull rise who is on the end of the drawbridge when the angle with measure x8 is 308? 458? 608?

SHORT RESPONSE Describe two ways to show that all isosceles right triangles are similar to each other.

PROVING THEOREM 7.8 Write a paragraph proof of the 458-458-908 Triangle Theorem. GIVEN c nDEF is a 458-458-908 triangle. PROVE c The hypotenuse is } 2 times as long as each leg. E D 458

EQUILATERAL TRIANGLE If an equilateral triangle has a side length of 20 inches, find the height of the triangle.

PROVING THEOREM 7.9 Write a paragraph proof of the 308-608-908 Triangle Theorem. GIVEN c nJKL is a 308-608-908 triangle. PROVE c The hypotenuse is twice as long as the shorter leg and the longer leg is } 3 times as long as the shorter leg. Plan for Proof Construct nJML congruent to nJKL. Then prove that nJKM is equilateral. Express the lengths of }JK and }JL in terms of x. 608 J L K M 3

MULTI-STEP PROBLEM You are creating a quilt that will have a traditional flying geese border, as shown below. a. Find all the angle measures of the small blue triangles and the large orange triangles. b. The width of the border is to be 3 inches. To create the large triangle, you cut a square of fabric in half. Not counting any extra fabric needed for seams, what size square do you need? c. What size square do you need to create each small triangle?

EXTENDED RESPONSE Use the figure at the right. You can use the fact that the converses of the 458-458-908 Triangle Theorem and the 308-608-908 Triangle Theorem are true. a. Find the values of r, s, t, u, v, and w. Explain the procedure you used to find the values. b. Which of the triangles, if any, is a 458-458-908 triangle? Explain. c. Which of the triangles, if any, is a 308-608-908 triangle? Explain. 1 w v u s t r 1 1

CHALLENGE In quadrilateral QRST, mR 5 608, mT 5 908, QR 5 RS, ST 5 8, TQ 5 8, and }RT and }QS intersect at point Z. a. Draw a diagram. b. Explain why nRQT nRST. c. Which is longer, QS or RT? Explain.

In the diagram, ] BD is the perpendicular bisector of}AC . (p. 303) 2x 1 6 16 B C D A x 1 7 2 Which pairs of segment lengths are equal?

In the diagram, ] BD is the perpendicular bisector of}AC . (p. 303) 2x 1 6 16 B C D A x 1 7 2 What is the value of x?

In the diagram, ] BD is the perpendicular bisector of}AC . (p. 303) 2x 1 6 16 B C D A x 1 7 2 Find CD.

Is it possible to build a triangle using the given side lengths? (p. 328) 4, 4, and 7

Is it possible to build a triangle using the given side lengths? (p. 328) 3, 3, and 9} 2

Is it possible to build a triangle using the given side lengths? (p. 328) 7, 15, and 21

Tell whether the given side lengths form a right triangle. (p. 441) 21, 22, and 5} 37

Tell whether the given side lengths form a right triangle. (p. 441) 3 }2 , 2, and 5 }2

Tell whether the given side lengths form a right triangle. (p. 441) 8, 10, and 14

VOCABULARY Copy and complete: The tangent ratio compares the length of ? to the length of ? .

WRITING Explain how you know that all right triangles with an acute angle measuring n8 are similar to each other.

FINDING TANGENT RATIOS Find tan A and tan B. Write each answer as a fraction and as a decimal rounded to four places. A 24 25

FINDING TANGENT RATIOS Find tan A and tan B. Write each answer as a fraction and as a decimal rounded to four places. B C A 35 37 12

FINDING TANGENT RATIOS Find tan A and tan B. Write each answer as a fraction and as a decimal rounded to four places. A B C 52 2

FINDING LEG LENGTHS Find the value of x to the nearest tenth. 12 418 x

FINDING LEG LENGTHS Find the value of x to the nearest tenth. 15 278 x

FINDING LEG LENGTHS Find the value of x to the nearest tenth. 22 588

FINDING LEG LENGTHS Find the value of x using the definition of tangent. Then find the value of x using the 458-458-908 Theorem or the 308-608-908 Theorem. Compare the results. 58 x 6 2

FINDING LEG LENGTHS Find the value of x using the definition of tangent. Then find the value of x using the 458-458-908 Theorem or the 308-608-908 Theorem. Compare the results. x 10 3

FINDING LEG LENGTHS Find the value of x using the definition of tangent. Then find the value of x using the 458-458-908 Theorem or the 308-608-908 Theorem. Compare the results. 608 4

SPECIAL RIGHT TRIANGLES Find tan 308 and tan 458 using the 458-458-908 Triangle Theorem and the 308-608-908 Triangle Theorem.

ERROR ANALYSIS Describe the error in the statement of the tangent ratio. Correct the statement, if possible. Otherwise, write not possible. tan D 5 18 }82 E F D 80 82

ERROR ANALYSIS Describe the error in the statement of the tangent ratio. Correct the statement, if possible. Otherwise, write not possible. A C B 30 55

WRITING Describe what you must know about a triangle in order to use the tangent ratio.

MULTIPLE CHOICE Which expression can be used to find the value of x in the triangle shown? A x 5 20 p tan 408 B x 5 } tan 408 20 C x 5 } 20 tan 408 D x 5 } 20 tan 508 x 408 20

MULTIPLE CHOICE What is the approximate value of x in the triangle shown? A 0.4 B 2.7 C 7.5 D 19.2 328 12

FINDING LEG LENGTHS Use a tangent ratio to find the value of x. Round to the nearest tenth. Check your solution using the tangent of the other acute angle. x 258 8

FINDING LEG LENGTHS Use a tangent ratio to find the value of x. Round to the nearest tenth. Check your solution using the tangent of the other acute angle. 408 1

FINDING LEG LENGTHS Use a tangent ratio to find the value of x. Round to the nearest tenth. Check your solution using the tangent of the other acute angle. 658 9 x

FINDING AREA Find the area of the triangle. Round to the nearest tenth. x 11 3

FINDING AREA Find the area of the triangle. Round to the nearest tenth. x 16 5

FINDING AREA Find the area of the triangle. Round to the nearest tenth. 7 228

FINDING PERIMETER Find the perimeter of the triangle. Round to the nearest tenth. 448 2

FINDING PERIMETER Find the perimeter of the triangle. Round to the nearest tenth. 318 6

FINDING PERIMETER Find the perimeter of the triangle. Round to the nearest tenth. 688 15

FINDING LENGTHS Find y. Then find z. Round to the nearest tenth. 428 120 y

FINDING LENGTHS Find y. Then find z. Round to the nearest tenth. 308 408 z 150 y

FINDING LENGTHS Find y. Then find z. Round to the nearest tenth. 458 288 82 z

CHALLENGE Find the perimeter of the figure at the right, where AC 5 26, AD 5 BF, and D is the midpoint of }AC A E D C G F H B 508 358

WASHINGTON MONUMENT A surveyor is standing 118 feet from the base of the Washington Monument. The surveyor measures the angle between the ground and the top of the monument to be 788. Find the height h of the Washington Monument to the nearest foot.

ROLLER COASTERS A roller coaster makes an angle of 528 with the ground. The horizontal distance from the crest of the hill to the bottom of the hill is about 121 feet, as shown. Find the height h of the roller coaster to the nearest foot. 121 ft h 5

CLASS PICTURE Use this information and diagram for Exercises 33 and 34. Your class is having a class picture taken on the lawn. The photographer is positioned 14 feet away from the center of the class. If she looks toward either end of the class, she turns 508. ISOSCELES TRIANGLE What is the distance between the ends of the class?

CLASS PICTURE Use this information and diagram for Exercises 33 and 34. Your class is having a class picture taken on the lawn. The photographer is positioned 14 feet away from the center of the class. If she looks toward either end of the class, she turns 508. MULTI-STEP PROBLEM The photographer wants to estimate how many more students can fit at the end of the first row. The photographer turns 508 to see the last student and another 108 to see the end of the camera range. a. Find the distance from the center to the last student in the row. b. Find the distance from the center to the end of the camera range. c. Use the results of parts (a) and (b) to estimate the length of the empty space. d. If each student needs 2 feet of space, about how many more students can fit at the end of the first row? Explain your reasoning.

SHORT RESPONSE Write expressions for the tangent of each acute angle in the triangle. Explain how the tangent of one acute angle is related to the tangent of the other acute angle. What kind of angle pair are A and B? B C A c

EYE CHART You are looking at an eye chart that is 20 feet away. Your eyes are level with the bottom of the E on the chart. To see the top of the E, you look up 18. How tall is the E? 18 20 ft

EXTENDED RESPONSE According to the Americans with Disabilities Act, a ramp cannot have an incline that is greater than 58. The regulations also state that the maximum rise of a ramp is 30 inches. When a ramp needs to reach a height greater than 30 inches, a series of ramps connected by 60 inch landings can be used, as shown below. a. What is the maximum horizontal length of the base of one ramp, in feet? Round to the nearest foot. b. If a doorway is 7.5 feet above the ground, what is the least number of ramps and landings you will need to lead to the doorway? Draw and label a diagram to justify your answer. c. To the nearest foot, what is the total length of the base of the system of ramps and landings in part (b)?

CHALLENGE The road salt shown is stored in a cone-shaped pile. The base of the cone has a circumference of 80 feet. The cone rises at an angle of 328. Find the height h of the cone. Then find the length s of the cone-shaped pile. 28 h s

The expressions given represent the angle measures of a triangle. Find the measure of each angle. Then classify the triangle by its angles. (p. 217) mA 5 x8 mB 5 4x8 mC 5 4x8

The expressions given represent the angle measures of a triangle. Find the measure of each angle. Then classify the triangle by its angles. (p. 217) mA 5 x8 mB 5 x8 mC 5 (5x 2 60)8

The expressions given represent the angle measures of a triangle. Find the measure of each angle. Then classify the triangle by its angles. (p. 217) mA 5 (x 1 20)8 mB 5 (3x 1 15)8 mC 5 (x 2 30)8

Copy and complete the statement with <, >, or 5. Explain. (p. 335) m 1 ? m 2 2 1 18 2

Copy and complete the statement with <, >, or 5. Explain. (p. 335) m 1 ? m 2 2 1

Copy and complete the statement with <, >, or 5. Explain. (p. 335) m 1 ? m 2 1 30 27

Find the unknown side length of the right triangle. (p. 433) 18 x 24

Find the unknown side length of the right triangle. (p. 433) x 157

Find the unknown side length of the right triangle. (p. 433) x 13

VOCABULARY Copy and complete: The sine ratio compares the length of ? to the length of ? .

WRITING Explain how to tell which side of a right triangle is adjacent to an angle and which side is the hypotenuse.

FINDING SINE RATIOS Find sin D and sin E. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 15 9 12 E F

FINDING SINE RATIOS Find sin D and sin E. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 37 F D E

FINDING SINE RATIOS Find sin D and sin E. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 5 28 53 F E D

ERROR ANALYSIS Explain why the students statement is incorrect. Write a correct statement for the sine of the angle. in A 5 5 }13 C B A 12 13 5

FINDING COSINE RATIOS Find cos X and cos Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 5 Z X Y

FINDING COSINE RATIOS Find cos X and cos Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 15 8 17 Z Y

FINDING COSINE RATIOS Find cos X and cos Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 3 13 3 X 2

USING SINE AND COSINE RATIOS Use a sine or cosine ratio to find the value of each variable. Round decimals to the nearest tenth. 18 y 328 x

USING SINE AND COSINE RATIOS Use a sine or cosine ratio to find the value of each variable. Round decimals to the nearest tenth. 10 b a 488

USING SINE AND COSINE RATIOS Use a sine or cosine ratio to find the value of each variable. Round decimals to the nearest tenth. w 5 718 v

USING SINE AND COSINE RATIOS Use a sine or cosine ratio to find the value of each variable. Round decimals to the nearest tenth. 26 s r 438

USING SINE AND COSINE RATIOS Use a sine or cosine ratio to find the value of each variable. Round decimals to the nearest tenth. 34 648

USING SINE AND COSINE RATIOS Use a sine or cosine ratio to find the value of each variable. Round decimals to the nearest tenth. 8 508

SPECIAL RIGHT TRIANGLES Use the 458-458-908 Triangle Theorem to find the sine and cosine of a 458 angle.

WRITING Describe what you must know about a triangle in order to use the sine ratio and the cosine ratio.

MULTIPLE CHOICE In nPQR, which expression can be used to find PQ? A 10 p cos 298 B 10 p sin 298 C } 10 sin 298 D } 10 cos 298 P 298 10 R P

ALGEBRA Find the value of x. Round decimals to the nearest tenth.

ALGEBRA Find the value of x. Round decimals to the nearest tenth.

ALGEBRA Find the value of x. Round decimals to the nearest tenth.

FINDING SINE AND COSINE RATIOS Find the unknown side length. Then find sin X and cos X. Write each answer as a fraction in simplest form and as a decimal. Round to four decimal places, if necessary. 14 Y Z X 7

FINDING SINE AND COSINE RATIOS Find the unknown side length. Then find sin X and cos X. Write each answer as a fraction in simplest form and as a decimal. Round to four decimal places, if necessary. 8 2 Y X

FINDING SINE AND COSINE RATIOS Find the unknown side length. Then find sin X and cos X. Write each answer as a fraction in simplest form and as a decimal. Round to four decimal places, if necessary. 35 12 Z

FINDING SINE AND COSINE RATIOS Find the unknown side length. Then find sin X and cos X. Write each answer as a fraction in simplest form and as a decimal. Round to four decimal places, if necessary. 6 3 5 Z X Y

FINDING SINE AND COSINE RATIOS Find the unknown side length. Then find sin X and cos X. Write each answer as a fraction in simplest form and as a decimal. Round to four decimal places, if necessary. 30 16 Z Y X

FINDING SINE AND COSINE RATIOS Find the unknown side length. Then find sin X and cos X. Write each answer as a fraction in simplest form and as a decimal. Round to four decimal places, if necessary. 65 56 Y Z X

ANGLE MEASURE Make a prediction about how you could use trigonometric ratios to find angle measures in a triangle.

MULTIPLE CHOICE In nJKL, m L 5 908. Which statement about nJKL cannot be true? A sin J 5 0.5 B sin J 5 0.1071 C sin J 5 0.8660 D sin J 5 1.1

PERIMETER Find the approximate perimeter of the figure. 558 1.5 cm 608 558

PERIMETER Find the approximate perimeter of the figure. .2 cm 1.5 cm 1.5 cm

CHALLENGE Let A be any acute angle of a right triangle. Show that (a) tan A 5 } sin A cos A and (b) (sin A) 2 1 (cos A) 2 5 1.

AIRPLANE RAMP The airplane door is 19 feet off the ground and the ramp has a 318 angle of elevation. What is the length y of the ramp? 318 19 ft

BLEACHERS Find the horizontal distance h the bleachers cover. Round to the nearest foot. 27 h ft 18 ft

SHORT RESPONSE You are flying a kite with 20 feet of string extended. The angle of elevation from the spool of string to the kite is 418. a. Draw and label a diagram to represent the situation. b. How far off the ground is the kite if you hold the spool 5 feet off the ground? Describe how the height where you hold the spool affects the height of the kite.

MULTI-STEP PROBLEM You want to hang a banner that is 29 feet tall from the third floor of your school. You need to know how tall the wall is, but there is a large bush in your way. a. You throw a 38 foot rope out of the window to your friend. She extends it to the end and measures the angle of elevation to be 708. How high is the window? b. The bush is 6 feet tall. Will your banner fit above the bush? c. What If? Suppose you need to find how far from the school your friend needs to stand. Which trigonometric ratio should you use? 38 ft 70

SHORT RESPONSE Nick uses the equation sin 498 5 x }16 to find BC in nABC. Tim uses the equation cos 418 5 x }16 . Which equation produces the correct answer? Explain. 16 C A B 498

TECHNOLOGY Use geometry drawing software to construct an angle. Mark three points on one side of the angle and construct segments perpendicular to that side at the points. Measure the legs of each triangle and calculate the sine of the angle. Is the sine the same for each triangle?

MULTIPLE REPRESENTATIONS You are standing on a cliff 30 feet above an ocean. You see a sailboat on the ocean. a. Drawing a Diagram Draw and label a diagram of the situation. b. Making a Table Make a table showing the angle of depression and the length of your line of sight. Use the angles 408, 508, 608, 708, and 808. c. Drawing a Graph Graph the values you found in part (b), with the angle measures on the x-axis. d. Making a Prediction Predict the length of the line of sight when the angle of depression is 308.

ALGEBRA If nEQU is equilateral and nRGT is a right triangle with RG 5 2, RT 5 1, and m T 5 908, show that sin E 5 cos G.

CHALLENGE Make a conjecture about the relationship between sine and cosine values. a. Make a table that gives the sine and cosine values for the acute angles of a 458-458-908 triangle, a 308-608-908 triangle, a 348-568-908 triangle, and a 178-738-908 triangle. b. Compare the sine and cosine values. What pattern(s) do you notice? c. Make a conjecture about the sine and cosine values in part (b). d. Is the conjecture in part (c) true for right triangles that are not special right triangles? Explain.

Rewrite the equation so that x is a function of y. (p. 877) y 5 } x

Rewrite the equation so that x is a function of y. (p. 877) y 5 3x 2 10

Rewrite the equation so that x is a function of y. (p. 877) y 5 x }9

Copy and complete the table. (p. 884) } x ? 0 ? 1 ? } 2 ? 2 ? 4

Copy and complete the table. (p. 884) x 1 }x ? 1 ? 1 }2 ? 3 ? 2 }7 ? 7

Copy and complete the table. (p. 884) x 2 }7 x 1 4 ? 0 ? 2 ? 6 ? 8 ? 10

Find the values of x and y in the triangle at the right. (p. 449) x y C D A B

VOCABULARY Copy and complete: To solve a right triangle means to find the measures of all of its ? and ? .

WRITING Explain when to use a trigonometric ratio to find a side length of a right triangle and when to use the Pythagorean Theorem.

USING INVERSE TANGENTS Use a calculator to approximate the measure of A to the nearest tenth of a degree. 18 12 A C

USING INVERSE TANGENTS Use a calculator to approximate the measure of A to the nearest tenth of a degree. 22 10 A C B

USING INVERSE TANGENTS Use a calculator to approximate the measure of A to the nearest tenth of a degree. 4 4 B A C

USING INVERSE SINES AND COSINES Use a calculator to approximate the measure of A to the nearest tenth of a degree. 5 11 C A

USING INVERSE SINES AND COSINES Use a calculator to approximate the measure of A to the nearest tenth of a degree. 6 10 A C B

USING INVERSE SINES AND COSINES Use a calculator to approximate the measure of A to the nearest tenth of a degree. 7 12 B C

MULTIPLE CHOICE Which expression is correct? A sin 21 JL }JK 5 m J B tan 21 } KL JL 5 m J C cos 21 JL }JK 5 m K D sin 21 JL }KL 5 m K J K L

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. 8 L K M 408

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. 10 P P N 6

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. R S 578

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. 12 9 A C B

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. 9 3 F D E

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. 14 16 H J G

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. 5.2 C B A 43.68

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. E D 8 3 14 3

SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimal answers to the nearest tenth. G H 10 7 8 29.9

ERROR ANALYSIS Describe and correct the students error in using an inverse trigonometric ratio. in21 } 7 WY 5 368 7 X W Y 36

ERROR ANALYSIS Describe and correct the students error in using an inverse trigonometric ratio. cos21 8 }15 5 m T V T 17 U 15 8

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. sin A 5 0.5

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. sin A 5 0.75

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. cos A 5 0.33

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. cos A 5 0.64

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. tan A 5 1.0

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. tan A 5 0.28

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. sin A 5 0.19

CALCULATOR Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. cos A 5 0.81

MULTIPLE CHOICE Which additional information would not be enough to solve nPRQ? A m P and PR B m P and m R C PQ and PR D m P and PQ

WRITING Explain why it is incorrect to say that tan21 x 5 } 1 tan x .

SPECIAL RIGHT TRIANGLES If sin A 5 1 }2 } 2 , what is m A? If sin B 5 1 }2 } 3 , what is m B?

TRIGONOMETRIC VALUES Use the Table of Trigonometric Ratios on page 925 to answer the questions. a. What angles have nearly the same sine and tangent values? b. What angle has the greatest difference in its sine and tangent value? c. What angle has a tangent value that is double its sine value? d. Is sin 2x equal to 2 p sin x?

CHALLENGE The perimeter of rectangle ABCD is 16 centimeters, and the ratio of its width to its length is 1 : 3. Segment BD divides the rectangle into two congruent triangles. Find the side lengths and angle measures of one of these triangles.

SOCCER A soccer ball is placed 10 feet away from the goal, which is 8 feet high. You kick the ball and it hits the crossbar along the top of the goal. What is the angle of elevation of your kick?

SHORT RESPONSE You are standing on a footbridge in a city park that is 12 feet high above a pond. You look down and see a duck in the water 7 feet away from the footbridge. What is the angle of depression? Explain your reasoning. 12 ft 7 ft duck y

CLAY In order to unload clay easily, the body of a dump truck must be elevated to at least 558. If the body of the dump truck is 14 feet long and has been raised 10 feet, will the clay pour out easily?

REASONING For nABC shown, each of the expressions sin 21 } BC AB , cos 21 } AC AB , and tan 21 } BC AC can be used to approximate the measure of A. Which expression would you choose? Explain your choice. 22 15 A B

MULTI-STEP PROBLEM You are standing on a plateau that is 800 feet above a basin where you can see two hikers. a. If the angle of depression from your line of sight to the hiker at B is 258, how far is the hiker from the base of the plateau? b. If the angle of depression from your line of sight to the hiker at C is 158, how far is the hiker from the base of the plateau? c. How far apart are the two hikers? Explain. A B C

MULTIPLE REPRESENTATIONS A local ranch offers trail rides to the public. It has a variety of different sized saddles to meet the needs of horse and rider. You are going to build saddle racks that are 11 inches high. To save wood, you decide to make each rack fit each saddle. a. Making a Table The lengths of the saddles range from 20 inches to 27 inches. Make a table showing the saddle rack length x and the measure of the adjacent angle y8. b. Drawing a Graph Use your table to draw a scatterplot. c. Making a Conjecture Make a conjecture about the relationship between the length of the rack and the angle needed. IN

OPEN-ENDED MATH Describe a real-world problem you could solve using a trigonometric ratio.

EXTENDED RESPONSE Your town is building a wind generator to create electricity for your school. The builder wants your geometry class to make sure that the guy wires are placed so that the tower is secure. By safety guidelines, the distance along the ground from the tower to the guy wires connection with the ground should be between 50% to 75% of the height of the guy wires connection with the tower. a. The tower is 64 feet tall. The builders plan to have the distance along the ground from the tower to the guy wires connection with the ground be 60% of the height of the tower. How far apart are the tower and the ground connection of the wire? b. How long will a guy wire need to be that is attached 60 feet above the ground? c. How long will a guy wire need to be that is attached 30 feet above the ground? d. Find the angle of elevation of each wire. Are the right triangles formed by the ground, tower, and wires congruent, similar, or neither? Explain. e. Explain which trigonometric ratios you used to solve the problem.

CHALLENGE Use the diagram of nABC. GIVEN c nABC with altitude }CD . PROVE c } sin A a 5 } sin B b C A D B a

Copy and complete the table. (p. 42) Number of sides Type of polygon 5 ? 12 ? ? Octagon ? Triangle 7 ? Number of sides Type of polygon ? n-gon ? Quadrilateral 10 ? 9 ? ? Hexagon

A point on an image and the transformation are given. Find the corresponding point on the original figure. (p. 272) Point on image: (5, 1); translation: (x, y) (x 1 3, y 2 2)

A point on an image and the transformation are given. Find the corresponding point on the original figure. (p. 272) Point on image: (4, 26); reflection: (x, y) (x, 2y)

A point on an image and the transformation are given. Find the corresponding point on the original figure. (p. 272) Point on image: (22, 3); translation: (x, y) (x 2 5, y 1 7)

Draw a dilation of the polygon with the given vertices using the given scale factor k. (p. 409) A(2, 2), B(21, 23), C(5, 23); k 5 2

Draw a dilation of the polygon with the given vertices using the given scale factor k. (p. 409) A(24, 22), B(22, 4), C(3, 6), D(6, 3); k 5 1 }2

Copy and complete: A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation ? .

WRITING What does it mean to solve a right triangle? What do you need to know to solve a right triangle?

WRITING Describe the difference between an angle of depression and an angle of elevation.

Find the unknown side length x. 6 x

Find the unknown side length x. 10 x 6

Find the unknown side length x. 12 x 369

Classify the triangle formed by the side lengths as acute, right, or obtuse. 6, 8, 9

Classify the triangle formed by the side lengths as acute, right, or obtuse. 4, 2, 5

Classify the triangle formed by the side lengths as acute, right, or obtuse. 10, 2} 2 , 6} 3

Classify the triangle formed by the side lengths as acute, right, or obtuse. 15, 20, 15

Classify the triangle formed by the side lengths as acute, right, or obtuse. 3, 3, 3} 2

Classify the triangle formed by the side lengths as acute, right, or obtuse. 13, 18, 3} 55

Find the value of x. 6 x 9

Find the value of x. x 9 4

Find the value of x. 6 4

Find the value of x. 5 2 x

Find the value of x. 12 16 x

Find the value of x. 20 25 x

Find the value of x. Write your answer in simplest radical form. x 6 6

Find the value of x. Write your answer in simplest radical form. x 14 3

Find the value of x. Write your answer in simplest radical form. 8 3 608

In Exercises 22 and 23, use the diagram. The angle between the bottom of a fence and the top of a tree is 758. The tree is 4 feet from the fence. How tall is the tree? Round your answer to the nearest foot. 758 4 ft

In Exercises 22 and 23, use the diagram. In Exercise 22, how tall is the tree if the angle is 558? 758 4 ft

Find the value of x to the nearest tenth. 548 x 3

Find the value of x to the nearest tenth. 258 x 20

Find the value of x to the nearest tenth. 388 10 x

Find sin X and cos X. Write each answer as a fraction, and as a decimal. Round to four decimals places, if necessary. 4 3 Y X Z

Find sin X and cos X. Write each answer as a fraction, and as a decimal. Round to four decimals places, if necessary. 10 Y Z X 149

Find sin X and cos X. Write each answer as a fraction, and as a decimal. Round to four decimals places, if necessary. 55 73 48 X Z Y

Solve the right triangle. Round decimal answers to the nearest tenth. 15 C 10 A B

Solve the right triangle. Round decimal answers to the nearest tenth. N 37

Solve the right triangle. Round decimal answers to the nearest tenth. 18 25 Y Z X

Find the measures of GED, GEF, and EFG. Find the lengths of }EG , }DF , }EF 08 D GF E 1