Question

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

Look at picture for reference

Answer 1

In a 30°-60°-90° right triangle, the length of the longer leg is √3 times the length of the shorter leg. So the length of the shorter leg is 8/√3, or (8/3)√3. This length is the hypotenuse of the isosceles right triangle. In an isosceles right triangle, the length of the hypotenuse is √2 times the length of a leg, so

x = ((8√3)/(3√2))(√2/√2) = (8/6)√6 = (4/3)√6