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Special Right Triangles

Special Right Triangles. Lesson 7-3. Investigation. In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle.

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Special Right Triangles

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  1. Special Right Triangles Lesson 7-3

  2. Investigation In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle. This triangle is also referred to as a 45-45-90 right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles.

  3. Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.

  4. Special Right Triangle Theorem 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg. Verify….

  5. Finding the Length of a Leg • Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.

  6. Example A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.

  7. Finding the Length of a Leg • The distance from one corner to the opposite corner of a square playground is 96 ft. To the nearest foot, how long is each side of the playground?

  8. Investigation The second special right triangle is the 30-60-90 right triangle, which is half of an equilateral triangle. Let’s start by using a little deductive reasoning to reveal a useful relationship in 30-60-90 right triangles.

  9. Investigation Triangle ABC is equilateral, and segment CD is an altitude. • What are m<A and m<B? • What are m<ADC and m<BDC? • What are m<ACD and m<BCD? • Is ΔADC = ΔBDC? Why? • Is AD=BD? Why? ~

  10. Find the length of the indicated side in each right triangle by using the conjecture you just made.

  11. Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.

  12. Special Right Triangle Theorem 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

  13. 30 8 60 Finding the Length of the Legs • Find the lengths of the legs of a 30°-60°-90° triangle with hypotenuse of length 8.

  14. 30 4Ö3 60 Finding the Length of the Legs • Find the lengths of the legs of a 30°-60°-90° triangle with hypotenuse of length 4Ö3.

  15. Using the Length of a Leg • The longer leg of a 30°-60°-90° triangle has length 18. Find the lengths of the shorter leg and the hypotenuse.

  16. Two Special* Right Triangles *what’s so special about them?

  17. Example Find the value of each variable. Write your answer in simplest radical form.

  18. Example Find the value of each variable. Write your answer in simplest radical form.

  19. Example What is the area of an equilateral triangle with a side length of 4 cm? 4 cm 4 cm 4 cm

  20. Finding Area

  21. Challenge! Find all side lengths

  22. Classwork • P. 369 • 1-20, 21-29 odd, 34-40

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