Question

find the length of side $x$ in simplest radical form with a rational denominator.

Explanation:

Step1: Identify triangle type

This is a 45-45-90 right triangle, so the legs are equal. The given leg length is 9, so the other leg is also 9.

Step2: Apply Pythagorean theorem

Use $c^2 = a^2 + b^2$, where $a=b=9$, $c=x$.
$$x^2 = 9^2 + 9^2$$

Step3: Calculate squared length

Compute the sum of the squared legs.
$$x^2 = 81 + 81 = 162$$

Step4: Solve for x, rationalize denominator

Take the square root and simplify.
$$x = \sqrt{162} = \sqrt{81 \times 2} = 9\sqrt{2}$$

Answer:

$9\sqrt{2}$