Question

the triangle below is isosceles. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Identify triangle type

It's an isosceles right triangle, so legs are equal (both 3), hypotenuse is \( x \).

Step2: Apply Pythagorean theorem

For a right triangle, \( a^2 + b^2 = c^2 \). Here, \( a = 3 \), \( b = 3 \), \( c = x \). So \( 3^2 + 3^2 = x^2 \).

Step3: Calculate \( x^2 \)

\( 9 + 9 = x^2 \) → \( x^2 = 18 \).

Step4: Simplify radical

\( x = \sqrt{18} = \sqrt{9\times2} = 3\sqrt{2} \) (rational denominator already, as \( \sqrt{2} \) is simplified).

Answer:

$3\sqrt{2}$