Question

the triangle below is isosceles. find the length of side x in simplest radical form with a rational denominator. triangle image with right angle and base 10, hypotenuse x

Explanation:

Step1: Identify triangle type

It's an isosceles right triangle (right angle + isosceles), so legs are equal. One leg is 10, so the other leg is also 10.

Step2: Apply Pythagorean theorem

For a right triangle, \( a^2 + b^2 = c^2 \). Here, \( a = 10 \), \( b = 10 \), \( c = x \). So \( 10^2 + 10^2 = x^2 \).

Step3: Calculate \( x^2 \)

\( 100 + 100 = x^2 \) → \( x^2 = 200 \).

Step4: Simplify radical

\( x = \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \) (denominator is rational, so this is simplest form).

Answer:

\( 10\sqrt{2} \)