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Chapter 7 Lesson 3

Chapter 7 Lesson 3. Objective: To use the properties of 45°-45°-90° triangles. Theorem 7-8: 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent and the length of the hypotenuse is √2 times the length of a leg. Hypotenuse = √2 •leg. 45°. hypotenuse. leg.

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Chapter 7 Lesson 3

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  1. Chapter 7 Lesson 3 Objective: To use the properties of 45°-45°-90° triangles.

  2. Theorem 7-8:45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent and the length of the hypotenuse is √2 times the length of a leg. Hypotenuse = √2•leg 45° hypotenuse leg 45° leg 45° x√2 x 45° x

  3. 45° 45° 45° 45° Example 1:Finding the Length of the Hypotenuse Find the value of each variable. a. b. 9 2√2 x h h=√2•9 h=9√2 x=√2•2√2 x=4

  4. 45° 45° Example 2:Finding the Length of the Hypotenuse Find the length of the hypotenuse of a 45°-45°-90° triangle with legs of length 5√3. 5√3 5√3 x=√2•5√3 x=5√6 x

  5. 45° 45° Example 3:Finding the Length of the Hypotenuse Find the length of the hypotenuse of a 45°-45°-90° triangle with legs of length 5√6. 5√6 5√6 x=√2•5√6 x=5√12 x=5√(4•3) x=10√3 x

  6. 45° 45° Example 4:Finding the Length of a Leg Find the value of x. 6=√2•x 6 x

  7. 45° 45° Example 5:Finding the Length of a Leg Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 10. 10=√2•x 10 x

  8. 45° 45° Example 6:Finding the Length of a Leg Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22. 22=√2•x 22 x

  9. 30° 30° 60° 60° Theorem 7-9:30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is √3 times the length of the shorter leg. hypotenuse = 2 • shorter leg longer leg = √3 • shorter leg hypotenuse long leg 2x x√3 short leg x

  10. 60° 8 x 30° y Example 7:Finding the Lengths of the Legs Find the value of each variable. Shorter Leg hypotenuse = 2 • shorter leg 8 = 2x x = 4 Longer Leg longer leg = √3 • shorter leg y = x√3 y = 4√3

  11. 60° 12 x 30° y Example 8:Finding the Lengths of the Legs Find the lengths of a 30°-60°-90° triangle with hypotenuse of length 12. Shorter Leg hypotenuse = 2 • shorter leg 12 = 2x x = 6 Longer Leg longer leg = √3 • shorter leg y = x√3 y = 6√3

  12. 60° 4√3 x 30° y Example 9:Finding the Lengths of the Legs Find the lengths of a 30°-60°-90° triangle with hypotenuse of length 4√3. Shorter Leg hypotenuse = 2 • shorter leg 4√3 = 2x x = 2√3 Longer Leg longer leg = √3 • shorter leg y = x√3 y = 2√3•√3 Y=6

  13. 30° 60° Example 10:Using the Length of a Leg Find the value of each variable. Shorter Leg long leg = √3 • short leg Hypotenuse Hyp. = 2 • shorter leg 5 x y

  14. 30° 60° Example 11:Using the Length of a Leg The shorter leg of a 30°-60°-90° has length √6. What are the lengths of the other sides? Leave your answers in simplest radical form. Longer Leg longer leg = √3 • shorter leg Hypotenuse hyp. = 2 • shorter leg x √6 y

  15. 30° 60° Example 12:Using the Length of a Leg The longer leg of a 30°-60°-90° has length 18. Find the length of the shorter leg and the hypotenuse. Shorter Leg long leg = √3 • short leg Hypotenuse hyp. = 2 • shorter leg 18 x y

  16. Assignment

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