Similar Right Triangles

Transcription

1 9.1 Similar Right Triangles Goals p Solve problems involving similar right triangles formed b the altitude drawn to the hpotenuse of a right triangle. p Use a geometric mean to solve problems. THEOREM 9.1 If the altitude is drawn to the hpotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. A TCBD STABC, TACD STABC, and TCBD STACD C D B Eample 1 Finding the Height of a Ramp Ramp Height A ramp has a cross section that is a right triangle. The diagram shows the approimate dimensions of this cross section. Find the height h of the ramp. K 4.7 ft h J M 11 ft 11.7 ft L B Theorem 9.1, T JKL STKML. Use similar triangles to write a proportion. KM K L Corresponding side lengths are in proportion. JK JL h Substitute h 11 ( 4.7 ) Cross product propert h 4.4 Solve for h. Answer The height of the ramp is about 4.4 feet. Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 187

2 Checkpoint Write similarit statements for the three triangles in the diagram. Then find the given length. Round decimals to the nearest tenth, if necessar. 1. Find AD. 2. Find NQ. 24 C N 10 Q 26 A D 18 B M P Sample answer: Sample answer: TABC STCBD STACD; TMNP STQNM STQMP; GEOMETRIC MEAN THEOREMS THEOREM 9.2 C In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. A BD CD D CD AD B THEOREM 9.3 In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hpotenuse and the segment of the hpotenuse that is adjacent to the leg. A B CB A B AC CB DB AC AD Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 188

3 Eample 2 Using a Geometric Mean Find the value of each variable. a. b a. Appl Theorem 9.2. b. Appl Theorem Checkpoint Find the value of the variable t t z Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 189

4 9.2 The Pthagorean Theorem Goals p Prove the Pthagorean Theorem. p Use the Pthagorean Theorem to solve problems. VOCABULARY Pthagorean triple A Pthagorean triple is a set of three positive integers a, b, and c that satisf the equation c 2 a 2 b 2. THEOREM 9.4: PYTHAGOREAN THEOREM In a right triangle, the square of the length of the hpotenuse is equal to the sum of the squares of the lengths of the legs. c 2 a 2 b 2 a c b Eample 1 Finding the Length of a Hpotenuse Find the length of the hpotenuse of the right triangle. Tell whether the side lengths form a Pthagorean triple. 8 (hpotenuse) 2 (leg) 2 (leg) 2 Pthagorean Theorem Substitute Multipl Add. 17 Find the positive square root. Answer The length of the hpotenuse is 17. Because the side lengths 8, 15, and 17 are integers, the form a Pthagorean triple. 15 Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 190

5 Eample 2 Finding the Length of a Leg Find the length of the leg of the right triangle (hpotenuse) 2 (leg) 2 (leg) 2 Pthagorean Theorem Substitute Multipl Subtract 225 from each side. 175 Find the positive square root. 25 p 7 Use product propert. 5 7 Simplif the radical. Answer The length of the leg is 5 7. Checkpoint Find the value of. Simplif answers that are radicals. Then tell whether the side lengths form a Pthagorean triple ; No 7; Yes Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 191

6 Eample 3 Finding the Area of a Triangle Find the area of the triangle to the nearest tenth of a square inch. 12 in. h 12 in. You are given that the base of the triangle is 18 inches, but ou do not know the height h. Because the triangle is isosceles, it can be divided into two congruent right angles with the given dimensions. Use the Pthagorean Theorem to find the value of h h 2 Pthagorean Theorem h 2 Multipl. 63 h 2 Subtract 81 from each side. 63 h Find the positive square root. 3 7 h Simplif the radical. Now find the area of the original triangle. 18 in. h 12 in. 9 in. A 1 2 bh Area of a triangle 3 7 in. 1 ( 18 )( 3 7 ) Substitute in Use a calculator. Answer The area of the triangle is about 71.4 square inches. Checkpoint Find the area of the triangle. Round our answer to the nearest tenth cm 9 cm 9 cm 10 ft 21 ft 39.6 cm ft 2 Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 192

7 9.3 The Converse of the Pthagorean Theorem Goals p Use the Converse of the Pthagorean Theorem to solve problems. p Use side lengths to classif triangles b their angle measures. THEOREM 9.5: CONVERSE OF THE PYTHAGOREAN THEOREM If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. B a C b c A If c 2 a 2 b 2,then TABC is a right triangle. Eample 1 Verifing Right Triangles Tell whether the triangle at the right is a right triangle. Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisf the equation c 2 a 2 b 2. ( 8 15 ) 2 ( 8 6 ) p ( 15 ) p ( 6 ) p p Answer The triangle is a right triangle Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 193

8 Checkpoint Tell whether the triangle is a right triangle The triangle is a right triangle. The triangle is not a right triangle. THEOREM 9.6 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. A b c If c 2 < a 2 b 2,then TABC is acute. C a B THEOREM 9.7 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. If c 2 > a 2 b 2,then TABC is obtuse. A b C c a B Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 194

9 Eample 2 Classifing Triangles Decide whether the set of numbers can represent the side lengths of a triangle. If the can, classif the triangle as right, acute, or obtuse. a. 28, 40, 48 b. 5.7, 12.2, 13.9 Compare the square of the length of the longest side with the sum of the squares of the lengths of the two shorter sides. a. c 2? a 2 b 2 Compare c 2 with a 2 b ? Substitute. 2304? Multipl < 2384 c 2 is less than a 2 b 2. Answer Because c 2 < a 2 b 2, the triangle is acute. b. c 2? a 2 b 2 Compare c 2 with a 2 b ? Substitute ? Multipl > c 2 is greater than a 2 b 2. Answer Because c 2 > a 2 b 2, the triangle is obtuse. Checkpoint Can the numbers represent the side lengths of a triangle? If so, classif the triangle as right, acute, or obtuse , 30, , 13, , 9, 12 es; right no es; obtuse Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 195

10 9.4 Special Right Triangles Goals p Find the side lengths of special right triangles. p Use special right triangles to solve real-life problems. VOCABULARY Special right triangles Special right triangles are right triangles whose angle measures are or THEOREM 9.8: TRIANGLE THEOREM In a triangle, the hpotenuse is 2 times as long as each leg. Hpotenuse 2 p leg THEOREM 9.9: TRIANGLE THEOREM In a triangle, the hpotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Hpotenuse 2 p shorter leg Longer leg 3 p shorter leg Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 196

11 Eample 1 Finding the Hpotenuse in a Triangle Find the value of. B the Triangle Sum Theorem, the measure of the third angle is 45. The triangle is a right triangle, so the length of the hpotenuse is 2 times the length of a leg Hpotenuse 2 p leg Triangle Theorem 2 p 7 Substitute. 7 2 Simplif. Eample 2 Side Lengths in a Triangle Find the values of s and t. Because the triangle is a triangle, the longer leg is 3 times the length of the shorter leg t 60 s Longer leg 3 p shorter leg Triangle Theorem 9 3 p s Substitute. 9 s Divide each side b p s s Simplif. Multipl numerator and denominator b 3. The length of the hpotenuse is twice the length of the shorter leg. Hpotenuse 2 p shorter leg Triangle Theorem t 2 p 3 3 Substitute. t 6 3 Simplif. Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 197

12 Checkpoint Find the values of the variables a b n m a 10 m b 20 n 30 Eample 3 Finding the Area of a Window Construction The window is a square. Find the area of the window. First find the side length s of the window b dividing it into two triangles. The length of the hpotenuse is 36 2 inches. Use this length to find s in p s Triangle Theorem 36 s Divide each side b 2. Use s 36 to find the area of the window. A s 2 Area of a square 36 2 Substitute Multipl. Answer The area of the window is 1296 square inches. Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 198

13 9.5 Trigonometric Ratios Goals p Find the sine, the cosine, and the tangent of an acute angle. p Use trigonometric ratios to solve real-life problems. VOCABULARY Trigonometric ratio A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. Sine A sine is a trigonometric ratio, abbreviated as sin. Cosine A cosine is a trigonometric ratio, abbreviated as cos. Tangent A tangent is a trigonometric ratio, abbreviated as tan. Angle of elevation An angle of elevation is the angle that our line of sight makes with a horizontal line when ou stand and look up at a point in the distance. TRIGONOMETRIC RATIOS Let TABC be a right triangle. The sine, the cosine, and the tangent of acute aa are defined as follows. sin A side opposite aa a B hpotenuse hpotenuse c cos A side adjacent to aa hpotenuse b c side opposite aa tan A a side adjacent to aa b c a A b C side adjacent to aa side opposite aa Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 199

14 Eample 1 Finding Trigonometric Ratios Find the sine, the cosine, and the tangent of ap. The length of the hpotenuse is 17. The R 15 length of the side opposite ap is 15, and the length of the side adjacent to ap is 8. opposite sin P 1 5 h potenuse adjacent 8 cos P h potenuse 1 7 tan P o pposite adjacent 8 P 8 17 Q Eample 2 Trigonometric Ratios for 60 Find the sine, the cosine, and the tangent of 60. Begin b sketching a triangle as shown at the right. To make the calculations simple, choose 1 as the length of the shorter leg. From the Triangle Theorem, it follows that the length of the longer leg is 3 and the length of the hpotenuse is 2. Label these lengths in the diagram. sin 60 o pp hp. 2 adj. cos h p. 2 tan 60 o pp adj Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 200

15 Checkpoint Use the diagram at the right to find the trigonometric ratio. 1. sin X X 12 Z Y 2. cos X tan Y Eample 3 Indirect Measurement Flag Pole You are measuring the height of a flag pole. You stand 19 feet from the base of the pole. You measure the angle of elevation from a point on the ground to the top of the pole to be 64. Estimate the height of the pole. opposite tan 64 a djacent Write trigonometric ratio. h tan Substitute. 19 tan 64 h Multipl each side b ( ) h Evaluate tan h Simplif. Answer The height of the flag pole is about 39 feet ft h Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 201

16 9.6 Solving Right Triangles Goals p Solve a right triangle. p Use right triangles to solve real-life problems. VOCABULARY Solve a right triangle To solve a right triangle means to determine the measures of all three angles and the lengths of all three sides. Eample 1 Solving a Right Triangle Solve the right triangle. Round decimals to the nearest tenth. B c Use the Pthagorean Theorem to find the length of 8 the hpotenuse c. (hpotenuse) 2 (leg) 2 (leg) 2 Pthagorean Theorem A 4 C c Substitute. c 2 80 Simplif. c 4 5 Find the positive square root. c 8.9 Use a calculator to approimate. Use a calculator to find the measure of ab. ( 4 8 ) 2nd TAN 26.6 aa and ab are complementar. The sum of their measures is 90. maa mab 90 aa and ab are complementar. maa Substitute for mab. maa 63.4 Subtract. Answer The side lengths are 4, 8, and 8.9. The angle measures are 26.6, 63.4, and 90. Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 202

17 Eample 2 Solving a Right Triangle Solve the right triangle. Round decimals to the nearest tenth. Use trigonometric ratios to find the values of p and q. P 30 q 53 Q p R sin Q o pp. hp. adj. cos Q h p. sin 53 q 30 cos 53 p sin 53 q 30 cos 53 p 30 ( ) q 30 ( ) p 24.0 q 18.1 p ap and aq are complementar. The sum of their measures is 90. map maq 90 ap and aq are complementar. map Substitute for maq. map 37 Subtract. Answer The side lengths of the triangle are 18.1, 24.0, and 30. The angle measures are 37, 53, and 90. Checkpoint Solve the right triangle. Round decimals to the nearest tenth. 1. C a A 89 B D e F f 8 31 E Side lengths: 39, 80, 89 Side lengths: 4.8, 8, 9.3 Angle measures: 26.0, Angle measures: 31, 59, 64.0, Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 203

18 Eample 3 Solving a Right Triangle Sports When a hocke plaer is 35 feet from the goal line, he shoots the puck directl at the goal. The angle of elevation at which the puck leaves the ice is 7. The height of the goal is 4 feet. Will the plaer score a goal? Begin b finding the height h of the puck at the goal line. Use a trigonometric ratio. tan 7 o pp. adj. Write trigonometric ratio. 4 ft 7 35 ft tan 7 h 35 Substitute. 35 tan 7 h Multipl each side b ( ) h Use a calculator. 4.3 h Multipl. Answer Because the height of the puck at the goal line ( 4.3 feet) is greater than the height of the goal (4 feet), the plaer will not score a goal. Checkpoint Complete the following eercise. 3. A hocke plaer is 27 feet from the goal line. He shoots the puck directl at the goal. The height of the goal is 4 feet. What is the maimum angle of elevation at which the plaer can shoot the puck and still score a goal? 4 ft 27 ft 8.4 Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 204

19 9.7 Vectors Goals p Find the magnitude and direction of a vector. p Add vectors. VOCABULARY Magnitude of a vector The magnitude of a vector is the distance from the initial point to the terminal point. Direction of a vector The direction of a vector is determined b the angle that the vector makes with a horizontal line. Equal vectors Two vectors are equal when the have the same magnitude and direction. Parallel vectors Two vectors are parallel when the have the same or opposite direction. Sum of vectors The sum of two vectors is a vector that joins the initial point of the first vector and the terminal point of the second vector. Eample 1 Finding the Magnitude of a Vector P( 4, 3) and Q(2, 1) are the initial and terminal points of a vector. Draw PQ &) in a coordinate plane. Then find its magnitude. Component form 2 1, 2 1 PQ &) 2 ( 4), 1 3 6, 4 Use the Distance Formula to find the magnitude. P Q PQ &) [ 2 )] ( 4 ( ) Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 205

20 Eample 2 Describing the Direction of a Vector The vector CD &) describes the velocit of a moving hot air balloon. The scale on each ais is in miles per hour. a. Find the speed of the balloon. b. Find the direction it is traveling relative to east. a. The magnitude of the vector CD &) represents the balloon s speed. Use the Distance Formula. CD &) (30 5) 2 (25 5) Answer The speed of the balloon is about 32 miles per hour. b. The tangent of the angle formed b N D the vector and a line drawn parallel to the -ais is ,or 0.8. Use a 25? 5 calculator to find the angle measure. C 0.8 2nd TAN 38.7 Answer The balloon is traveling in a direction about 38.7 north of east. W W 5 N S S C D E E Checkpoint Complete the following eercise. 1. The vector represents the velocit of a moving hot air balloon. The scale on each ais is in miles per hour. Find the balloon s speed and direction relative to west. W J N 5 5 E 25.5 miles per hour; 78.7 south of west K S Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 206

21 Eample 3 Identifing Equal and Parallel Vectors In the diagram, these vectors have the same direction: AB ****), CD &*), IJ **). These vectors are equal: AB ****), IJ **). These vectors are parallel: AB ****), CD &*), GH &*), IJ **). A B D E G C 1 3 I H F J ADDING VECTORS Sum of Two Vectors The sum of u*) a 1, b 1 and v*) a 2, b 2 is u*) v*) a 1 a 2, b 1 b 2. Eample 4 Finding the Sum of Two Vectors Let u*) 6, 2 and v*) 8, 7. Find the sum u*) v*). 5 To find the sum, add the horizontal components and add the vertical components of u and v. u v v 3 5 u u*) v*) 6 ( 8), 2 7 2, 5 Checkpoint Find the sum of the vectors. 2. 6, 0, 1, , 2, 7, 6 7, 3 2, 4 Copright McDougal Littell/Houghton Mifflin Compan All rights reserved. Chapter 9 Geometr Notetaking Guide 207