There are three right triangles in this figure. Set up Pythagorean theorem for each of them.
Let y be the altitude of the largest triangle.
Let z be the left side.
We have for large triangle:
x² + z² = (9+3)²
For left sub-triangle:
y² + 3² = z²
For right sub-triangle:
y² + 81 = x²
So the three equations are
x² + z² = 144
y² + 9 = z²
y² + 81 = x²
Solve last equation for y², plug into second equation, solve that for z² in terms of x², and lastly plug z² into the first equation to find
x² + x² - 72 = 144
2x² = 216
x² = 108
x = √108 = √(36·3) = 6√3
The answer is C) 6√3
