Question

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

Explanation:

Step1: Use the property of isosceles right - triangle

In an isosceles right - triangle (a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle), the ratio of the sides is \(1:1:\sqrt{2}\). Let the legs be of length \(x\) (the two non - hypotenuse sides) and the hypotenuse be \(c\). The formula is \(c = x\sqrt{2}\). Here, \(c = 10\).

Step2: Solve for \(x\)

We have the equation \(10=x\sqrt{2}\). To solve for \(x\), we use the formula \(x=\frac{c}{\sqrt{2}}\). Substitute \(c = 10\) into the formula: \(x=\frac{10}{\sqrt{2}}\).

Step3: Rationalize the denominator

Multiply the numerator and denominator by \(\sqrt{2}\): \(x=\frac{10\sqrt{2}}{\sqrt{2}\times\sqrt{2}}\). Since \(\sqrt{2}\times\sqrt{2}=2\), then \(x = 5\sqrt{2}\).

Answer:

\(5\sqrt{2}\)