You've been given a 30 - 60 - 90 triangle and a 45 - 45 - 90 triangle here. To solve it, focus on the left triangle and use that to solve the other.
In a 30 - 60 - 90 triangle, the pattern for the legs is as follows:
short leg: 1
long leg: √3
hypotenuse: 2
You're told the hypotenuse is 4√3, so the short leg is going to be half of that, 2√3. Notice in the pattern above above how the hypotenuse is double the short leg. (SIDE C)
Now that we have those two legs, we use the Pythagorean Theorem to get the long leg.
(4√3)² = x² + (2√3)²
48 = x² + 12
36 = x²
6 = x or -6 = x
We are working with lengths, so the -6 is discarded and that tells us the long leg is 6. (SIDE A)
Now to the 45 - 45 - 90 triangle, whose pattern is as follows.
Each leg = 1
Hypotenuse = √2
(Note this triangle is isosceles, so yes, the two legs are the same length).
Since it's a 45-45-90 triangle, we can buy one solved leg and get one free, and so each leg is 6. (SIDE D)
Using the pattern, we multiply our legs by √2, and the hypotenuse of the 45 - 45 - 90 triangle is 6√2. (SIDE B)
Let's put it all together.
Side A = 6, Side B = 6√2, SIde C = 2√3, Side D = 6 (the first choice in the bubbles).
