Question

Solve for x.

A) 10.5
B) 10
C) 6√3
D) 20

Answer 1

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

Answer 2

Applying the Pythagorean theorem to the three right triangles yields x = 3√6, as derived from the given equations. Here option C is correct.

The three right triangles in the figure can be described by the Pythagorean theorem as follows:

For the largest triangle:

x^2 + z^2 = (9 + 3)^2

For the left sub-triangle:

y^2 + 3^2 = z^2

For the right sub-triangle:

y^2 + 81 = x^2

Combining the three equations, we get:

x^2 + z^2 = 144

y^2 + 9 = z^2

y^2 + 81 = x^2

Solving the last equation for y^2 and substituting it into the second equation, we get:

9 + 81 = z^2

z^2 = 90

Substituting z^2 into the first equation, we find:

x^2 + 90 = 144

x^2 = 54

x = √54 = √(6 * 9) = 3√6

The correct answer is x = 3√6, which corresponds to option C.