Using the Pythagorean Theorem and Its Converse

Transcription

1 7 ig Idea 1 HPTR SUMMR IG IDS Using the Pythagorean Theorem and Its onverse For our Notebook The Pythagorean Theorem states that in a right triangle the square of the length of the hypotenuse c is equal to the sum of the squares of the lengths of the legs a and b, so that c 2 5 a 2 1 b 2. The onverse of the Pythagorean Theorem can be used to determine if a triangle is a right triangle. b a c b a c b c a If c 2 5 a 2 1 b 2, then m 5 90 and n is a right triangle. If c 2 < a 2 1 b 2, then m < 90 and n is an acute triangle. If c 2 > a 2 1 b 2, then m > 90 and n is an obtuse triangle. ig Idea 2 Using Special Relationships in Right Triangles GOMTRI MN In right n, altitude } D forms two smaller triangles so that nd, n D, n. lso, D } D 5 D } D, } 5 } D, and } 5 } D. D SPIL RIGHT TRINGLS Triangle Triangle hypotenuse 5 legpï } 2 hypotenuse 5 2p shorter leg longer leg 5 shorter legpï } 3 ig Idea 3 Using Trigonometric Ratios to Solve Right Triangles The tangent, sine, and cosine ratios can be used to find unknown side lengths and angle measures of right triangles. The values of tan, sin, and cos depend only on the angle measure and not on the side length. tan 5 opp. } adj. 5 } tan 21 } 5 m sin 5 opp. } hyp. 5 } sin 21 } 5 m cos 5 adj. } hyp. 5 } cos 21 } 5 m adjacent toa hypotenuse oppositea hapter Summary 493

2 7 HPTR RVIW K VOULR HPTR RVIW classzone.com Multi-Language Glossary Vocabulary practice For a list of postulates and theorems, see pp Pythagorean triple, p. 435 trigonometric ratio, p. 466 tangent, p. 466 sine, p. 473 cosine, p. 473 angle of elevation, p. 475 angle of depression, p. 475 solve a right triangle, p. 43 inverse tangent, p. 43 inverse sine, p. 43 inverse cosine, p. 43 VOULR RISS 1. opy and complete: Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation?. 2. WRITING What does it mean to solve a right triangle? What do you need to know to solve a right triangle? 3. WRITING Describe the difference between an angle of depression and an angle of elevation. RVIW MPLS ND RISS Use the review eamples and eercises below to check your understanding of the concepts you have learned in each lesson of hapter pply the Pythagorean Theorem pp M P L Find the value of. ecause is the length of the hypotenuse of a right triangle, you can use the Pythagorean Theorem to find its value. (hypotenuse) 2 5 (leg) 2 1 (leg) Pythagorean Theorem Substitute. Simplify. Find the positive square root MPLS 1 and 2 on pp for s. 4 6 RISS Find the unknown side length hapter 7 Right Triangles and Trigonometry

3 classzone.com hapter Review Practice 7.2 Use the onverse of the Pythagorean Theorem pp M P L Tell whether the given triangle is a right triangle. heck to see whether the side lengths satisfy the equation c 2 5 a 2 1 b Ï } < 146 The triangle is not a right triangle. It is an acute triangle MPL 2 on p. 442 for s RISS lassify the triangle formed by the side lengths as acute, right, or obtuse. 7. 6,, 9. 4, 2, , 2Ï } 2, 6Ï } , 20, , 3, 3Ï } , 1, 3Ï } Use Similar Right Triangles pp M P L Find the value of. y Theorem 7.6, you know that 4 is the geometric mean of and 2. } } 2 Write a proportion ross Products Property Divide. MPLS 2 and 3 on pp for s RISS Find the value of hapter Review 495

4 7 Special 7.4 HPTR RVIW Right Triangles pp M P L Find the length of the hypotenuse. y the Triangle Sum Theorem, the measure of the third angle must be 45. Then the triangle is a triangle hypotenuse 5 legpï } Ï } Triangle Theorem Substitute. MPLS 1, 2, and 5 on pp for s RISS Find the value of. Write your answer in simplest radical form pply the Tangent Ratio pp M P L Find the value of. tan 37 5 opp. } adj. Write ratio for tangent of 37. tan 37 5 } Substitute. p tan 37 5 Multiply each side by ø Use a calculator to simplify. MPL 2 RISS In ercises 22 and 23, use the diagram. on p. 467 for s The angle between the bottom of a fence and the top of a tree is 75. The tree is 4 feet from the fence. How tall is the tree? Round your answer to the nearest foot. 23. In ercise 22, how tall is the tree if the angle is 55? Find the value of to the nearest tenth ft hapter 7 Right Triangles and Trigonometry

5 classzone.com hapter Review Practice 7.6 pply the Sine and osine Ratios pp M P L Find sin and sin. sin 5 opp. } hyp. 5 } 5 15 } 17 ø sin 5 opp. } hyp. 5 } 5 } 17 ø MPLS 1 and 2 on pp for s RISS Find sin and cos. Write each answer as a fraction, and as a decimal. Round to four decimals places, if necessary. 27. Z Z Z 7.7 Solve Right Triangles pp M P L Use a calculator to approimate the measure of to the nearest tenth of a degree. ecause tan 5 1 } } , tan m. Use a calculator to evaluate this epression. tan ø So, the measure of is approimately MPL 3 on p. 44 for s RISS Solve the right triangle. Round decimal answers to the nearest tenth N 6 M Z 1 10 L 33. Find the measures of GD, GF, and FG. Find the lengths of } G, } DF, } F D G F hapter Review 497

6 7 Find HPTR TST the value of. Write your answer in simplest radical form lassify the triangle as acute, right, or obtuse. 4. 5, 15, 5 Ï } , 6.7, , 7, Find the value of. Round decimal answers to the nearest tenth Find the value of each variable. Write your answer in simplest radical form y y y Solve the right triangle. Round decimal answers to the nearest tenth F 15. H D 9.2 G J 16. FLGPOL Julie is 6 feet tall. If she stands 15 feet from the flagpole and holds a cardboard square, the edges of the square line up with the top and bottom of the flagpole. pproimate the height of the flagpole. 17. HILLS The length of a hill in your neighborhood is 2000 feet. The height of the hill is 750 feet. What is the angle of elevation of the hill? 2000 ft a 750 ft 49 hapter 7 Right Triangles and Trigonometry

7 7 LGR RVIW GRPH ND SOLV QUDRTI QUTIONS lgebra classzone.com The graph of y 5 a 2 1 b 1 c is a parabola that opens upward if a > 0 and opens downward if a < 0. The -coordinate of the verte is 2 b } 2a. The ais of symmetry is the vertical line 5 2 b } 2a. M P L 1 Graph a quadratic function Graph the equation y ecause a 5 21 and 21 < 0, the graph opens downward. 1 y 5 2 The verte has -coordinate 2 b } 2a } 2(21) The y-coordinate of the verte is 2(2) 2 1 4(2) So, the verte is (2, 1) and the ais of symmetry is 5 2. Use a table of values to draw a parabola through the plotted points. M P L 2 Solve a quadratic equation by graphing Solve the equation Write the equation in the standard form a 2 1 b 1 c 5 0: Graph the related quadratic function y , as shown. The -intercepts of the graph are 21 and 3. So, the solutions of are 21 and 3. heck the solution algebraically. (21) 2 2 2(21) (3) 2 2 2(3) y RISS MPL 1 for s. 1 6 Graph the quadratic function. Label the verte and ais of symmetry. 1. y y y y y 5 1 } y MPL 2 for s. 7 1 Solve the quadratic equation by graphing. heck solutions algebraically } lgebra Review 499

8 7 Standardized TST PRPRTION MULTIPL HOI QUSTIONS If you have difficulty solving a multiple choice question directly, you may be able to use another approach to eliminate incorrect answer choices and obtain the correct answer. P RO L M 1 ou ride your bike at an average speed of 10 miles per hour. How long does it take you to ride one time around the triangular park shown in the diagram? 1.7 mi L 0.1 h 0.3 h 0.2 h D 0.4 h J 1.5 mi K MTHOD 1 SOLV DIRTL The park is a right triangle. Use the Pythagorean Theorem to find KL. Find the perimeter of njkl. Then find how long it takes to ride around the park. STP 1 Find KL. Use the Pythagorean Theorem. JK 2 1 KL 2 5 JL KL KL KL KL 5 0. STP 2 Find the perimeter of njkl. P 5 JK 1 JL 1 KL mi STP 3 Find the time t (in hours) it takes you to go around the park. Rate 3 Time 5 Distance (10 mi/h) p t 5 4 mi t h The correct answer is D. D MTHOD 2 LIMINT HOIS nother method is to find how far you can travel in the given times to eliminate choices that are not reasonable. STP 1 Find how far you will travel in each of the given times. Use the formula rt 5 d. hoice : 0.1(10) 5 1 mi hoice : 0.2(10) 5 2 mi hoice : 0.3(10) 5 3 mi hoice D: 0.4(10) 5 4 mi The distance around two sides of the park is mi. ut you need to travel around all three sides, which is longer. Since 1 < 3.2, 2 < 3.2, and 3 < 3.2. ou can eliminate choices,, and. STP 2 heck that D is the correct answer. If the distance around the park is 4 miles, then KL JK 2 JL mi. pply the onverse of the Pythagorean Theorem The correct answer is D. D 500 hapter 7 Right Triangles and Trigonometry

9 P RO L M 2 What is the height of nw? 4 4 Ï } 3 D Ï } 3 W MTHOD 1 SOLV DIRTL Draw altitude } Z to form two congruent triangles. MTHOD 2 LIMINT HOIS nother method is to use theorems about triangles to eliminate incorrect choices. Draw altitude } Z to form two congruent right triangles. h 60 W 4 Z 4? Let h be the length of the longer leg of nz. The length of the shorter leg is 4. longer leg 5 Ï } 3 p shorter leg h 5 4 Ï } 3 The correct answer is. D W 4 Z 4 onsider nzw. y the Triangle Inequality, W < WZ 1 Z. So, < 4 1 Z and Z > 4. ou can eliminate choice. lso, Z must be less than the hypotenuse of nwz. ou can eliminate choices and D. The correct answer is. D PRTI plain why you can eliminate the highlighted answer choice. 1. In the figure shown, what is the length of } F? 9 9 Ï } 2 1 D 9 Ï } 5 9 H G F 2. Which of the following lengths are side lengths of a right triangle? 2, 21, 23 3, 4, 5 9, 16, 1 D 11, 16, In npqr, PQ 5 QR 5 13 and PR What is the length of the altitude drawn from verte Q? D 13 Standardized Test Preparation 501

10 7 Standardized TST PRTI MULTIPL HOI 1. Which epression gives the correct length for W in the diagram below? Ï } Ï } 3 5 Ï } Ï } 2 D The area of nfg is 400 square meters. To the nearest tenth of a meter, what is the length of side } G? Z W 5. It takes 14 minutes to walk from your house to your friend s house on the path shown in red. If you walk at the same speed, about how many minutes will it take on the path shown in blue? your house 500 yd 6 minutes 10 minutes 700 yd friend s house minutes D 13 minutes F 10.0 meters 44.7 meters 20.0 meters D 56.7 meters 3. Which epression can be used to find the value of in the diagram below? m 17 tan 29 5 } 17 cos 29 5 } 17 G 6. Which equation can be used to find QR in the diagram below? QR } } 7 15 } 5 } QR QR QR 5 Ï } D P 15 ft S 7 ft P QR } } 15 R tan 61 5 } 17 D cos 61 5 } fire station, a police station, and a hospital are not positioned in a straight line. The distance from the police station to the fire station is 4 miles. The distance from the fire station to the hospital is 3 miles. Which of the following could not be the distance from the police station to the hospital? 7. Stitches are sewn along the black line segments in the potholder shown below. There are 10 stitches per inch. Which is the closest estimate of the number of stitches used? in. 1 mile 5 miles 2 miles D 6 miles D hapter 7 Right Triangles and Trigonometry

11 STT TST PRTI classzone.com GRIDDD NSWR. design on a T-shirt is made of a square and four equilateral triangles. The side length of the square is 4 inches. Find the distance (in inches) from point to point. Round to the nearest tenth. SHORT RSPONS 10. The diagram shows the side of a set of stairs. In the diagram, the smaller right triangles are congruent. plain how to find the lengths, y, and z. 11 in. in. y z 9. Use the diagram below. Find KM to the nearest tenth of a unit. 11. ou drive due north from Dalton to ristol. Net, you drive from ristol to Hilldale. Finally, you drive from Hilldale to Dalton. Is Hilldale due west of ristol? plain. L 12 9 Hilldale 71 mi ristol N K 11 N M W NW N 100 mi 74 mi SW S S Dalton TNDD RSPONS 12. The design for part of a water ride at an amusement park is shown. The ride carries people up a track along ramp }. Then riders travel down a water chute along ramp }. a. How high is the ride above point D? plain. b. What is the total distance from point to point to point? plain ft D 42 ft 13. formula for the area of a triangle is Heron s Formula. For a triangle with side lengths F, FG, and G, the formula is 5 Ï }}} s(s 2 F)(s 2 FG)(s 2 G), where s 5 1 } 2 (F 1 FG 1 G). F a. In nfg shown, F 5 FG 5 15, and G 5 1. Use Heron s formula to find the area of nfg. Round to the nearest tenth. b. Use the formula 5 1 } 2 bh to find the area of nfg. Round to the nearest tenth. c. Use Heron s formula to justify that the area of an equilateral triangle with side length is 5 2 } 4 Ï } 3. G Standardized Test Practice 503