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Special Right Triangles 5.1 (M2)

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Pythagorean Theorem

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Right Triangle Theorems 45 o -45 o -90 o Triangle Theorem Hypotenuse is times as long as each leg 30 o -60 o -90 o Triangle Theorem Hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg

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EXAMPLE 1 Find hypotenuse length in a 45-45-90 triangle o o o Find the length of the hypotenuse. a. SOLUTION hypotenuse = leg 2 = 8= 8 2 Substitute. 45-45-90 Triangle Theorem o o o By the Triangle Sum Theorem, the measure of the third angle must be 45 º. Then the triangle is a 45 º -45 º - 90 º triangle, so by Theorem 7.8, the hypotenuse is 2 times as long as each leg. a.

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EXAMPLE 1 Find hypotenuse length in a 45-45-90 triangle o o o hypotenuse = leg 2 Substitute. 45-45-90 Triangle Theorem o o o = 3 22 = 3 2 Product of square roots = 6 Simplify. b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. 45 - 45 - 90 o o o Find the length of the hypotenuse. b.

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EXAMPLE 2 Find leg lengths in a 45-45-90 triangle o o o Find the lengths of the legs in the triangle. SOLUTION By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. 45 - 45 - 90 o o o hypotenuse = leg 2 Substitute. 45-45-90 Triangle Theorem o o o 2 5 = x= x 2 2 5 2 = 2 x 2 5 = x Divide each side by 2 Simplify.

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EXAMPLE 3 Standardized Test Practice SOLUTION By the Corollary to the Triangle Sum Theorem, the triangle is a triangle. 45 - 45 - 90 o o o

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EXAMPLE 3 Standardized Test Practice hypotenuse = leg 2 Substitute. 45-45-90 Triangle Theorem o o o = 252 WX The correct answer is B.

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GUIDED PRACTICE for Examples 1, 2, and 3 Find the value of the variable. 1.2.3. ANSWER 2 2 8 2

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GUIDED PRACTICE for Examples 1, 2, and 3 4. Find the leg length of a 45°- 45°- 90° triangle with a hypotenuse length of 6. 3 2 ANSWER

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EXAMPLE 4 Find the height of an equilateral triangle Logo The logo on a recycling bin resembles an equilateral triangle with side lengths of 6 centimeters. What is the approximate height of the logo? SOLUTION Draw the equilateral triangle described. Its altitude forms the longer leg of two 30°-60°-60° triangles. The length h of the altitude is approximately the height of the logo. h = 3 5.2 cm 3 longer leg = shorter leg 3

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EXAMPLE 5 Find lengths in a 30-60-90 triangle o oo Find the values of x and y. Write your answer in simplest radical form. STEP 1 Find the value of x. longer leg = shorter leg 3 9 = x 3 9 3 = x 9 3 3 3 9 3 3 3 3 Simplify. Multiply fractions. Triangle Theorem 30 - 60 - 90 o o o Divide each side by 3 Multiply numerator and denominator by 3 Substitute.

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EXAMPLE 5 Find lengths in a 30-60-90 triangle o oo hypotenuse = 2 shorter leg STEP 2 Find the value of y. y = 2 3 = 6 3 3 Substitute and simplify. Triangle Theorem 30 - 60 - 90 o o o

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EXAMPLE 6 Find a height Dump Truck The body of a dump truck is raised to empty a load of sand. How high is the 14 foot body from the frame when it is tipped upward at the given angle? a. 45 angle o b.60 angle o SOLUTION When the body is raised 45 above the frame, the height h is the length of a leg of a triangle. The length of the hypotenuse is 14 feet. a. 45 - 45 - 90 o o o o

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EXAMPLE 6 Find a height 14 = h2 14 2 = h 9.9 h Triangle Theorem 45 - 45 - 90 o o o Divide each side by 2 Use a calculator to approximate. When the angle of elevation is 45, the body is about 9 feet 11 inches above the frame. o b. When the body is raised 60, the height h is the length of the longer leg of a triangle. The length of the hypotenuse is 14 feet. 30 - 60 - 90 o o o o

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EXAMPLE 6 Find a height hypotenuse = 2 shorter leg Triangle Theorem 30 - 60 - 90 o o o 14 = 2 s Substitute. 7 = s Divide each side by 2. longer leg = shorter leg 3 Triangle Theorem 30 - 60 - 90 o o o h = 7 3 Substitute. h 12.1 Use a calculator to approximate. When the angle of elevation is 60, the body is about 12 feet 1 inch above the frame. o

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GUIDED PRACTICE for Examples 4, 5, and 6 Find the value of the variable. ANSWER 3 3 2

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GUIDED PRACTICE for Examples 4, 5, and 6 What If? In Example 6, what is the height of the body of the dump truck if it is raised 30° above the frame? 7. ANSWER 7 ft SAMPLE ANSWER The shorter side is adjacent to the 60° angle, the longer side is adjacent to the 30° angle. In a 30°- 60°- 90° triangle, describe the location of the shorter side. Describe the location of the longer side? 8.

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