In the 30-60-90 triangle below, side s has a length of
and side q has a
length of

Step-by-step explanation:

The Pythagorean triple for a 30-60-90 triangle is, in terms of patterns,

(x, x√3, 2x) where x is the length of the side across from the 30 degree angle, x√3 is the length of the side across from the 60 degree angle, and the length of the hypotenuse is represented by 2x. We have the length of the hypotenuse given as 10. Therefore:

2x = 10 (the formula for the length of the hypotenuse is set equal to the value of the hypotenuse, allowing us to solve for x) so

x = 5. Look up above at the triple. The side length across from the 30 degree angle measures x; if x = 5, then side s = 5.

The formula for the side length across from the 60 degree angle is x√3, and again, if x = 5, side q = 5√3 which is choice C.