Question

Find the missing sides in this 30°-60°-90° right triangle. The hypotenuse is 10 and one of the shorter sides is 6. What is the length of the other shorter side and the longer side?

1) 4 and 6
2) 3 and 6
3) 4 and 8
4) 3 and 8

Answer 1

In a 30°-60°-90° right triangle, the longer leg is 6, and by using the ratios for such a triangle, the shorter leg is found to be 3. Thus, the lengths of the missing sides are 3 (shorter leg) and 6 (longer leg), making option 2) correct.

Step-by-step explanation:

The task is to find the lengths of the missing sides in a right triangle with a hypotenuse of 10 and a known side of 6. In a 30°-60°-90° right triangle, the sides are in a specific ratio. The side opposite the 30° angle (shorter leg) is half the hypotenuse, the side opposite the 60° angle (longer leg) is the shorter leg multiplied by √3, and the side opposite the 90° angle is the hypotenuse.

To find the missing sides, we use the provided 6 as either the shorter leg or the longer leg.

• If 6 is the shorter leg, then hypotenuse should be 2 × 6 = 12, which is not the case.
• If 6 is the longer leg, then the shorter leg is 6 / √3 = 3.46 (approximately 3 when rounded to the nearest whole number), and the hypotenuse is 2 × 3.46 = 6.93 × 2 = 10, which matches the hypotenuse given.

Therefore, the shorter leg is 3 and the longer leg is 6, making option 2) correct: 3 and 6.