Question

An equilateral triangle ABC and a semicircle are attached as shown. D is mid point of arc AB. if the height of the triangle is $2\sqrt{3}$, Then Perimeter of the figure is

• A. $2(\pi +4)$
• B. $2(\pi -4)$
• C. $2(4-\pi )$
• D. $(-4-\pi )\sqrt{3}$
  

Answer 1

The correct option is A $2(\pi +4)$

$ABC$ is an equilateral triangle.$D$ is the mid-point and divides $\angle C$ in to two parts.

$\therefore$The side lengths in ${30}^{\circ }-{60}^{\circ }-{90}^{\circ }$ are in the ratio $1:\sqrt{3}:2$

$\Rightarrow$Radius of the circle which is also the base of ${30}^{\circ }-{60}^{\circ }-{90}^{\circ }$ triangles, the height of the triangle and the side of the triangle are in the same ratio.

$\Rightarrow \frac{height}{\sqrt{3}}=\frac{side}{2}$

$\Rightarrow \frac{2\sqrt{3}}{\sqrt{3}}=\frac{side}{2}$

$\Rightarrow side=4$units

Since diameter of the semi-circle and the length of a side of the equilateral triangle are the same

Perimeter$=2\times side+\frac{\pi \left(side\right)}{2}$

$=2\times 4+\pi \frac{4}{2}=8+2\pi =2(4+\pi )$