Question

find x using special right triangles. write your answer in simplified radical form.

Explanation:

Step1: Identify the triangle type

This is a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle. In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Let the legs be of length \(a\) and the hypotenuse be \(c\). The formula is \(c = a\sqrt{2}\). Here, the hypotenuse \(c = 10\), and we need to find the leg \(x\) (since the non - hypotenuse sides are equal in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle).

Step2: Solve for \(x\)

From \(c=a\sqrt{2}\), we can rewrite it as \(a=\frac{c}{\sqrt{2}}\). Substitute \(c = 10\) into the formula: \(x=\frac{10}{\sqrt{2}}\). Rationalize the denominator: \(x=\frac{10\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{10\sqrt{2}}{2}=5\sqrt{2}\).

Answer:

\(5\sqrt{2}\)