7.4 Special Right Triangles

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1 7.4 Special Right Triangles Goal p Use the relationships among the sides in special right triangles. Your Notes The etended ratio of the side lengths of a triangle is 1:1: Ï 2. THEOREM 7.8: TRIANGLE THEOREM In a triangle, the hypotenuse is times as long as each leg. hypotenuse 5 leg p 2 Eample 1 Find hypotenuse length in a triangle Find the length of the hypotenuse. a. b Remember the following properties of radicals: Ï a p Ï b 5 Ï a p b ; Ï a p a 5 a a. By the Triangle Sum Theorem, the measure of the third angle must be. Then the triangle is a triangle, so by Theorem 7.8, the hypotenuse is times as long as each leg. hypotenuse 5 leg p Substitute. b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. hypotenuse 5 leg p p Substitute. 5 p Product of square roots 5 Simplify. 188 Lesson 7.4 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

2 7.4 Special Right Triangles Goal p Use the relationships among the sides in special right triangles. Your Notes The etended ratio of the side lengths of a triangle is 1:1: Ï 2. THEOREM 7.8: TRIANGLE THEOREM In a triangle, the hypotenuse is Ï 2 times as long as each leg. hypotenuse 5 leg p Ï 2 2 Eample 1 Find hypotenuse length in a triangle Find the length of the hypotenuse. a. b Remember the following properties of radicals: Ï a p Ï b 5 Ï a p b ; Ï a p a 5 a a. By the Triangle Sum Theorem, the measure of the third angle must be. Then the triangle is a triangle, so by Theorem 7.8, the hypotenuse is Ï 2 times as long as each leg. hypotenuse 5 leg p Ï Ï 2 Substitute. b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. hypotenuse 5 leg p Ï Ï 2 p Ï 2 Substitute. 5 4 p 2 Product of square roots 5 8 Simplify. 188 Lesson 7.4 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

3 Eample 2 Find leg lengths in a triangle Find the lengths of the legs in 9 2 the triangle. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. hypotenuse 5 leg p p Substitute. 5 Divide each side by. 5 Simplify. Checkpoint Find the value of the variable The etended ratio of the side lengths of a triangle is 1: Ï : 2. THEOREM 7.9: TRIANGLE THEOREM In a triangle, the hypotenuse is as long as the shorter leg, and the longer leg is times as long as the shorter leg. hypotenuse 5 p shorter leg 2 longer leg 5 shorter leg p 08 Copyright Holt McDougal. All rights reserved. Lesson 7.4 Geometry Notetaking Guide 189

4 Eample 2 Find leg lengths in a triangle Find the lengths of the legs in 9 2 the triangle. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. hypotenuse 5 leg p Ï Ï 2 5 p Ï 2 Substitute. 9 Ï 2 Ï 2 5 Ï 2 Ï Simplify. Divide each side by Ï 2. Checkpoint Find the value of the variable The etended ratio of the side lengths of a triangle is 1: Ï : 2. THEOREM 7.9: TRIANGLE THEOREM In a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is Ï times as long as the shorter leg. hypotenuse 5 2 p shorter leg 2 longer leg 5 shorter leg p Ï 08 Copyright Holt McDougal. All rights reserved. Lesson 7.4 Geometry Notetaking Guide 189

5 Remember that in an equilateral triangle, the altitude to a side is also the median to that side. So, altitude BD AC. Eample Find the height of an equilateral triangle Music You make a guitar pick that resembles an equilateral triangle with side lengths of 2 millimeters. What is the approimate height of the pick? Draw the equilateral triangle described. Its altitude forms the longer leg of two triangles. The length h of the altitude is approimately the height of the pick. longer leg 5 shorter leg p h 5 p < mm 2 mm A D h 2 mm 1 mm D 1 mm C Eample 4 Find lengths in a triangle Find the values of and y. Write your answer in simplest radical form. y 08 8 Step 1 Find the value of. longer leg 5 shorter leg p 5 Substitute. 5 Divide each side by. p 5 Multiply numerator and denominator by. 5 Multiply fractions. Step 2 Find the value of y. hypotenuse 5 p shorter leg y 5 p Lesson 7.4 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

6 Remember that in an equilateral triangle, the altitude to a side is also the median to that side. So, altitude BD bisects AC. Eample Find the height of an equilateral triangle Music You make a guitar pick that resembles an equilateral triangle with side lengths of 2 millimeters. What is the approimate height of the pick? Draw the equilateral triangle described. Its altitude forms the longer leg of two triangles. The length h of the altitude is approimately the height of the pick. longer leg 5 shorter leg p Ï h 5 1 p Ï < 27.7 mm 2 mm A D h 2 mm 1 mm D 1 mm C Eample 4 Find lengths in a triangle Find the values of and y. Write your answer in simplest radical form. y 08 8 Step 1 Find the value of. longer leg 5 shorter leg p Ï 8 Ï 8 Ï p Ï Ï 8 Ï 8 5 Ï Substitute. 5 Divide each side by Ï. 5 Multiply numerator and denominator by Ï. 5 Multiply fractions. Step 2 Find the value of y. hypotenuse 5 2 p shorter leg y 5 2 p 8 Ï 5 1 Ï 190 Lesson 7.4 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

7 Eample 5 Find a height Windshield wipers A car is turned off while the windshield wipers are moving. The 24 inch wipers stop, making a angle with the bottom of the windshield. How far from the bottom of the windshield are the ends of the wipers? The distance d is the length 24 in. of the longer leg of a triangle. The length of the hypotenuse is inches. hypotenuse 5 p shorter leg p s Substitute. 5 s Divide each side by. longer leg 5 shorter leg p d 5 d < The ends of the wipers are about bottom of the windshield. Substitute. Approimate. inches from the Checkpoint In Eercises and 4, find the value of the variable h Homework 5. In Eample 5, how far from the bottom of the windshield are the ends of the wipers if they make a 08 angle with the bottom of the windshield? Copyright Holt McDougal. All rights reserved. Lesson 7.4 Geometry Notetaking Guide 191

8 Eample 5 Find a height Windshield wipers A car is turned off while the windshield wipers are moving. The 24 inch wipers stop, making a angle with the bottom of the windshield. How far from the bottom of the windshield are the ends of the wipers? The distance d is the length 24 in. of the longer leg of a triangle. The length of the hypotenuse is 24 inches. hypotenuse 5 2 p shorter leg p s Substitute s Divide each side by 2. longer leg 5 shorter leg p Ï d 5 12 Ï Substitute. d < 20.8 Approimate. The ends of the wipers are about 20.8 inches from the bottom of the windshield. Checkpoint In Eercises and 4, find the value of the variable. Homework h 12 5 h 5 Ï 5. In Eample 5, how far from the bottom of the windshield are the ends of the wipers if they make a 08 angle with the bottom of the windshield? 12 inches Copyright Holt McDougal. All rights reserved. Lesson 7.4 Geometry Notetaking Guide 191

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