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Understanding 30-60-90 Right Triangles: Side Lengths and Theorems

A 30-60-90 triangle is a special right triangle characterized by its angles of 30° and 60°. In such triangles, the hypotenuse measures twice the length of the shorter leg, while the longer leg is equal to the shorter leg multiplied by √3. Recognizing these side lengths and angles is crucial for solving problems involving 30-60-90 triangles. This guide provides helpful hints and examples to illustrate the relationships between the sides and angles, including how to solve for unknown lengths.

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Understanding 30-60-90 Right Triangles: Side Lengths and Theorems

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  1. Special Right Triangles 30-60-90

  2. General 30°-60°-90° Triangle

  3. 30°-60°-90° Triangle Theorem • In a 30°-60°-90° triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times

  4. How do you know it is 30°-60°-90°? • You have to recognize the side lengths or angle measures • Example pictures: 60° 6 10 5

  5. Examples: Solve for x and y. 1. 2. 3.

  6. Helpful Hints • To go from the short leg of the triangle to the hypotenuse, multiply by 2 • To go from the short leg of the triangle to the long leg, multiply by

  7. Hints continued… • To go from the hypotenuse of the triangle to the short leg, divide by 2 • To go from the long leg of the triangle to the short leg, divide by • You have to RATIONALIZE after you divide by SO…multiply by and divide by 3

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