Question

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

image of a right isosceles triangle with hypotenuse 4 and legs ( x )

Explanation:

Step1: Identify triangle type

It's an isosceles right triangle, so legs are equal (\(x = \text{other leg}\)), hypotenuse \(c = 4\).

Step2: Apply Pythagorean theorem

For a right triangle, \(a^2 + b^2 = c^2\). Here, \(a = x\), \(b = x\), \(c = 4\). So \(x^2 + x^2 = 4^2\).

Step3: Simplify equation

Combine like terms: \(2x^2 = 16\). Divide both sides by 2: \(x^2 = 8\).

Step4: Solve for \(x\)

Take square root: \(x = \sqrt{8}\). Simplify: \(x = 2\sqrt{2}\) (rational denominator already, as \(\sqrt{8} = 2\sqrt{2}\)).

Answer:

\(2\sqrt{2}\)