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Special Right Triangles

Special Right Triangles. 5.1 (M2). Pythagorean Theorem. 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

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Special Right Triangles

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  1. Special Right Triangles 5.1 (M2)

  2. Pythagorean Theorem

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

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

  5. o o o Find hypotenuse length in a 45-45-90triangle b. o o o hypotenuse = leg 2 45-45-90 Triangle Theorem By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. b. 2 2 = 3 = 32 o o o 45 - 45 - 90 EXAMPLE 1 Find the length of the hypotenuse. Substitute. Product of square roots = 6 Simplify.

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

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

  8. hypotenuse = leg WX = 25 The correct answer is B. 2 2 EXAMPLE 3 Standardized Test Practice o o o 45-45-90 Triangle Theorem Substitute.

  9. 1. 2. 3. 2 2 2 ANSWER ANSWER 8 ANSWER for Examples 1, 2, and 3 GUIDED PRACTICE Find the value of the variable.

  10. 2 3 ANSWER for Examples 1, 2, and 3 GUIDED PRACTICE 4. Find the leg length of a 45°- 45°- 90° triangle with a hypotenuse length of 6.

  11. h = 35.2 cm 3 3 Draw the equilateral triangle described. Its altitude forms the longer leg of two 30°-60°-60° triangles. The length hof the altitude is approximately the height of the logo. longer leg = shorter leg 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

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

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

  14. 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? o o a. b. 60 angle 45 angle o a. When the body is raised 45 above the frame, the height his the length of a leg of a triangle. The length of the hypotenuse is 14 feet. o o o 45 - 45 - 90 EXAMPLE 6 Find a height Dump Truck SOLUTION

  15. 14 = h 14 = h Divide each side by 2 2 o When the angle of elevation is 45, the body is about 9feet 11 inches above the frame. 2 o b. 9.9 h When the body is raised 60, the height his the length of the longer leg of a triangle. The length of the hypotenuse is 14 feet. o o o o o o 45 - 45 - 90 30 - 60 - 90 EXAMPLE 6 Find a height Triangle Theorem Use a calculator to approximate.

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

  17. 3 ANSWER 3 ANSWER 2 for Examples 4, 5, and 6 GUIDED PRACTICE Find the value of the variable.

  18. 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. SAMPLE ANSWER 8. In a 30°- 60°- 90° triangle, describe the location of the shorter side. Describe the location of the longer side? The shorter side is adjacent to the 60° angle, the longer side is adjacent to the 30° angle. 7 ft ANSWER for Examples 4, 5, and 6 GUIDED PRACTICE

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