Final answer:
In a 30°-60°-90° triangle with a hypotenuse of 10 inches, the shorter leg measures 5 inches, and the longer leg measures 5√3 inches (approximately 8.66 inches). These lengths are determined by the specific side ratios of this type of triangle.
Step-by-step explanation:
The student is asking about the length of a leg in a special type of right triangle known as a 30°-60°-90° triangle. In such a triangle, the sides have a unique ratio. The length of the hypotenuse (the side opposite the 90° angle) is twice the length of the shorter leg (the side opposite the 30° angle). If the hypotenuse is 10 inches, then the shorter leg would be half that length, which is 5 inches. The longer leg (opposite the 60° angle) is √3 times the length of the shorter leg, which would be 5√3 inches or approximately 8.66 inches.
To find the length of the legs of the triangle, we use the defined ratios of a 30°-60°-90° triangle. Therefore, for a hypotenuse of 10 inches:
The shorter leg (opposite the 30°) = hypotenuse / 2 = 10 / 2 = 5 inches.
The longer leg (opposite the 60°) = shorter leg √3 = 5 √3 inches.
This relationship is based on the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c²) is equal to the sum of the squares of the lengths of the other two sides (a² + b²).