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Properties of 30°-60°-90° Triangles | Understanding Theorem 7-9

Learn how to apply Theorem 7-9 to find lengths in 30°-60°-90° triangles. Explore examples and solve for legs and hypotenuse lengths using ratios and trigonometry principles.

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Properties of 30°-60°-90° Triangles | Understanding Theorem 7-9

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  1. Chapter 7 Lesson 3 Objective: To use the properties of 30°-60°-90° triangle.

  2. 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

  3. 60° 8 x 30° y Example 1: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

  4. 60° 12 x 30° y Example 2: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

  5. 60° 4√3 x 30° y Example 3: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

  6. 30° 60° Example 4: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

  7. 30° 60° Example 5: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

  8. 30° 60° Example 6: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

  9. Homework Page 369 – 371 #12-22; 24-27; 30-33; 35;38

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