Question

solve for x. leave your answer in simplest radical form.

Explanation:

Step1: Find the height (h) of the right triangle on the right.

Let the height be \( h \). Using the Pythagorean theorem (\( a^2 + b^2 = c^2 \)) for the right triangle with legs \( h \) and \( 4 \), and hypotenuse \( 6 \):
\( h^2 + 4^2 = 6^2 \)
\( h^2 + 16 = 36 \)
\( h^2 = 36 - 16 = 20 \)

Step2: Use the height to find \( x \) in the left right triangle.

The left triangle has hypotenuse \( 8 \), height \( h \) (from Step 1, \( h^2 = 20 \)), and leg \( x \). Apply the Pythagorean theorem:
\( x^2 + h^2 = 8^2 \)
Substitute \( h^2 = 20 \):
\( x^2 + 20 = 64 \)
\( x^2 = 64 - 20 = 44 \)
Simplify \( \sqrt{44} \): \( \sqrt{4 \times 11} = 2\sqrt{11} \)

Answer:

\( 2\sqrt{11} \)