Question

Find the missing side lengths, put answer as radical in simplest form

Look at picture for reference

Answer 1

In an isosceles right triangle, the length of the hypotenuse is √2 times the length of a leg. So the length of the hypotenuse is 6√2, which is the length of the hypotenuse of the 30°-60°-90° triangle. In a 30°-60°-90° triangle, the length of the shorter leg is one-half the length of the hypotenuse, and the length of the longer leg is √3 times the length of the shorter leg. So the length of the shorter leg is 3√2, and it follows that x = 3√6.