Question

Find the perimeter of an isosceles right triangle with congruent sides measuring 7 cm.

Answer 1

First, we draw an isosceles right triangle with congruent side measuring 7 cm.

In ΔXYZ, $\angle$X = $\angle$Z as XY = YZ = 7 cm and $\angle$Y = 90°.
Then,

$\angle$X + $\angle$Y +$\angle$Z = 180°
$\Rightarrow$$\angle$X + 90° +$\angle$X = 180°
$\Rightarrow$ 2$\angle$X = 180° – 90° = 90°
$\Rightarrow$ $\angle$X =$\frac{90°}{2}=45°$
$\Rightarrow$ $\angle$X = $\angle$Z=45°

Thus, Δ XYZ is a 45°-45°-90° triangle.
By 45°-45°-90° triangle theorem, we get

Since XY = 7 cm,

$$\begin{gathered} \mathrm{XY}=\frac{1}{\sqrt{2}}\times \mathrm{XZ} \\ \Rightarrow XZ=7\sqrt{2} \end{gathered}$$
Thus, the perimeter of the isosceles triangle XYZ is
XY + YZ + ZX = 7 + 7 +$7\sqrt{2}$ = 14 +$7\sqrt{2}$ = 7(2 +$\sqrt{2}$ ) cm.