Question

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

Explanation:

Step1: Identify the triangle type

This is a \(45^{\circ}-45^{\circ}-90^{\circ}\) right - 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, one of the legs (let's assume the non - hypotenuse side adjacent to \(45^{\circ}\)) is \(4\sqrt{2}\), and we need to find the hypotenuse \(x\).

Step2: Apply the \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle ratio formula

We know that for a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, if the length of a leg \(a = 4\sqrt{2}\), and the hypotenuse \(x\) (using \(x=a\sqrt{2}\)). Substitute \(a = 4\sqrt{2}\) into the formula:

$$x=(4\sqrt{2})\times\sqrt{2}$$

$$x = 4\times(\sqrt{2}\times\sqrt{2})$$

Since \(\sqrt{2}\times\sqrt{2}=2\), then \(x = 4\times2\)

Answer:

\(8\)