Remark
The key to this problem is finding the length of the hypotenuse of the 45o - 45o - 90o triangle. After than you can use the tangent to find x
Step One
Find the hypotenuse of the 45o - 45o - 90o right triangle
Use the fact that in this triangle, the two smaller sides of the triangle are equal.
They have a value of x. Since they are given as 12 we can solve for the hypotenuse.
Formula
a^2 + b^2 = c^2
Givens
a = 12
b = 12
c = ?
Sub and solve.
12^2 + 12^2 = c^2 Take out the common factor of 12^2
c^2 = 12^2 * (1 + 1)
c^2 = 2* 12^2 Take the square root of both sides.
sqrt(c^2) = sqrt(2 * 12^2)
sqrt(c^2) = sqrt(2) * sqrt(12^2)
sqrt(c) = 12 sqrt(2)
Step Two
Now use the 30 - 60 - 90 triangle to solve for x
The opposite side to the 60o angle is 12 sqrt(2)
Tan(60) = opposite / adjacent
Tan(60) = 12 sqrt(2) / adjacent
Tan(60) = sqrt(3)
adjacent = 12*sqrt(2)/sqrt(3)
Rationalize the denominator
adjacent = 12*sqrt(2) * sqrt(3) / (sqrt(3) * sqrt(3))
adjacent = 12*sqrt(6) / 3
adjacent = 4 sqrt(6)
Answer : D
