Question

A right triangle has angles which measure $30,60$ and $90$ degrees. If the perimeter of this triangle is $15+$ $5\sqrt{3}$, then the length of the hypotenuse of this triangle is

• A. $5$
• B. $7.5$
• C. $10$
• D. $12.5$

Answer 1

The correct option is C $10$

Let the hypotenuse be $H$, perpendicular be $P$ and base be $B$.
Now, $\frac{P}{H}=\sin 30$
$\Rightarrow \frac{H}{2}=P$
And, $\frac{B}{H}=\cos 30$
$\Rightarrow B=\frac{\sqrt{3}H}{2}$
Now, perimeter $=$ $15+15\sqrt{3}$
$\Rightarrow H+\frac{H}{2}+\frac{\sqrt{3}H}{2}=15+5\sqrt{3}$
$\Rightarrow \frac{3H}{2}+\frac{\sqrt{3}H}{2}=15+5\sqrt{3}$
Now comparing, $\frac{3H}{2}=15$
$\Rightarrow H=10$ cm