Question

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

Explanation:

Step1: Identify equal sides

This is a right isosceles triangle, so the two legs are equal. The given leg is length 3, so the other leg (adjacent to the right angle) is also 3.

Step2: Apply Pythagorean theorem

For a right triangle, $a^2 + b^2 = c^2$, where $a=3$, $b=3$, $c=x$.
$$3^2 + 3^2 = x^2$$

Step3: Calculate left-hand side

Compute the sum of the squared legs.
$$9 + 9 = x^2 \implies 18 = x^2$$

Step4: Solve for x

Take the square root of both sides, simplify the radical.
$$x = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$

Answer:

$3\sqrt{2}$