Question

Right triangle XYZ hasm∠Y = 90° , m∠X = 45° and side XZ = 36Find the area of triangle XYZ.

Answer 1

Given:

In a right triangle XYZ,

$\begin{gathered} m\angle Y=90\degree \\ m\angle X=45\degree \\ \text{The side, }XZ=36 \end{gathered}$

To find the area of the triangle:

Let us find the base and height of the triangle.

Using the trigonometric ratio,

$\begin{gathered} \sin \theta=\frac{Opp}{Hyp} \\ \sin 45^{\circ}=\frac{YZ}{XZ} \\ \frac{1}{\sqrt[]{2}}=\frac{YZ}{36} \\ YZ=\frac{36}{\sqrt[]{2}} \\ =\frac{36}{\sqrt[]{2}}\times\frac{\sqrt[]{2}}{\sqrt[]{2}} \\ =\frac{36\sqrt[]{2}}{2} \\ YZ=18\sqrt[]{2}\ldots\ldots\ldots(1) \end{gathered}$

Using the trigonometric ratio,

$\begin{gathered} \cos \theta=\frac{\text{Adj}}{Hyp} \\ \cos 45^{\circ}=\frac{XY}{XZ} \\ \frac{1}{\sqrt[]{2}}=\frac{XY}{36} \\ XY=\frac{36}{\sqrt[]{2}} \\ =\frac{36}{\sqrt[]{2}}\times\frac{\sqrt[]{2}}{\sqrt[]{2}} \\ =\frac{36\sqrt[]{2}}{2} \\ XY=18\sqrt[]{2}\ldots\ldots\ldots(2) \end{gathered}$

The formula of area of the triangle is,

$\begin{gathered} A=\frac{1}{2}\times base\times height \\ =\frac{1}{2}\times XY\times YZ \\ =\frac{1}{2}\times18\sqrt[]{2}\times18\sqrt[]{2} \\ =324 \end{gathered}$

Hence, the area of the triangle is 324 square units.