Question

What are the values of the side lengths of a triangle with a 30-degree, 60-degree, and 90-degree angles, where the short leg equals 13?

(a) 13, 13√3, 26
(b) 13√3, 26, 13
(c) 13, 26, 13√3
(d) 26, 13, 13√3

Answer 1

The side lengths of a 30°-60°-90° right triangle, where the short leg is 13, are short leg = 13, long leg = 13√3, and hypotenuse = 26, which corresponds to option (a) 13, 13√3, 26.

Step-by-step explanation:

The triangle described is a 30°-60°-90° right triangle, which is a special type of right triangle. The lengths of the sides of such a triangle are in a specific ratio. For any 30°-60°-90° triangle, the length of the hypotenuse (the side opposite the 90-degree angle) is twice that of the short leg (the side opposite the 30-degree angle), and the length of the long leg (the side opposite the 60-degree angle) is √3 times the length of the short leg. Given that the short leg equals 13, we can calculate the other lengths of the triangle.

The hypotenuse is twice the short leg: 2 x 13 = 26.
The long leg is √3 times the short leg: 13 x (√3) = 13√3.

Therefore, the values of the side lengths of the triangle are: short leg = 13, long leg = 13√3, and hypotenuse = 26. The correct answer is (a) 13, 13√3, 26.