Question

find the value of each variable.
11.
12.
14.
15.
17.
18.
20.
22.

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 ratio of the sides is \(1:1:\sqrt{2}\). Let the legs be \(x\) and \(y\) (since the non - hypotenuse angles are \(45^{\circ}\), the legs are equal), and the hypotenuse \(c=\sqrt{10}\).

Step2: Use the side - ratio formula

The formula for the sides of a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle is \(c = x\sqrt{2}\) (where \(c\) is the hypotenuse and \(x,y\) are the legs). Given \(c=\sqrt{10}\), we solve for \(x\) (and \(y\) since \(x = y\)):

$$x=\frac{c}{\sqrt{2}}$$

Substitute \(c = \sqrt{10}\) into the formula:

$$x=\frac{\sqrt{10}}{\sqrt{2}}=\sqrt{\frac{10}{2}}=\sqrt{5}$$

Since \(x = y\) in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, \(y=\sqrt{5}\)

Answer:

\(x = \sqrt{5}\), \(y=\sqrt{5}\)