Question

the triangle below is isosceles. find the length of side ( x ) in simplest radical form with a rational denominator.
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attempt 1 out of 2
( x = )
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Explanation:

Step1: Identify equal sides

Since it's an isosceles right triangle, the two legs are equal: the leg labeled $x$ and the unlabeled leg are the same length. The hypotenuse is 9.

Step2: Apply Pythagorean theorem

For a right triangle, $a^2 + b^2 = c^2$. Here, $a=x$, $b=x$, $c=9$.
$$x^2 + x^2 = 9^2$$

Step3: Simplify the equation

Combine like terms and calculate the square:
$$2x^2 = 81$$

Step4: Solve for $x^2$

Divide both sides by 2:
$$x^2 = \frac{81}{2}$$

Step5: Solve for $x$ and rationalize

Take the square root, then rationalize the denominator:
$$x = \sqrt{\frac{81}{2}} = \frac{9}{\sqrt{2}} = \frac{9\sqrt{2}}{2}$$

Answer:

$\frac{9\sqrt{2}}{2}$