5.2.4 Applying Special Right Triangles (continued)
The 30 ° -60 ° -90 ° Triangle Theorem is used here to determine the unknown lengths of the legs of a triangle. The given triangle is a 30 ° -60 ° -90 ° triangle. The length of the hypotenuse is given as 14 units. Begin by expressing the unknown length of the short leg in terms of the known length of the hypotenuse. This yields a length for the shorter leg of 7. The length of the longer leg is the length of the shorter leg times 3, or 7 3.
Example 4
The 30 ° -60 ° -90 ° Triangle Theorem is used to determine the unknown length of the long leg of a triangle in this application example. Is the height of the equilateral triangle less than the allowed space of 50 inches? The line segment that defines the height divides the triangle into two congruent 30 ° -60 ° -90 ° triangles. The length of the hypotenuse for these triangles, the length of the side of the equilateral triangle, is given as 54 inches. According to the 30 ° -60 ° -90 ° Triangle Theorem, the length of the short leg is half the length of the hypotenuse, or 54/2 = 27 inches. By the 30 ° -60 ° -90 ° Triangle Theorem, the length of the longer leg is 27 3. Use a calculator to find the approximate length of the longer leg, also the height of the original equilateral triangle, to be approximately 46.8 inches. This is less than the available wall height, so the painting will fit within the space.
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