Question

The area of a regular polygon is $75\sqrt{3}c{m}^2$. Find the length of OM.O is the center of the polygon.

• A. 8 cm
• B. 4 cm
• C. 2 cm
• D. 5 cm

Answer 1

The correct option is D 5 cm

$△ABC$ is an equilateral $△$

$\therefore Area=\frac{\sqrt{3}}{4}{a}^2=75\sqrt{3}⟹$ $a^2=300$

$a=10\sqrt{3}$cm

$AM=MC=\frac{10\sqrt{3}}{2}=5\sqrt{3}$cm

Area = $\frac{1}{2}\times AC\times BM$

$75\sqrt{3}=\frac{1}{2}\times 10\sqrt{3}\times BM$

$\therefore BM=15$cm

$△AOC,△BOC$ and $△AOB$ are of equal area

$\therefore ar\left(△AOC\right)=\frac{1}{3}ar\left(△ABC\right)=\frac{1}{3}\times 75\sqrt{3}=25\sqrt{3}c{m}^2$

In $△AOC$,

$ar\left(△AOC\right)=\frac{1}{2}\times AC\times OM$

$\Rightarrow 25\sqrt{3}=\frac{1}{2}\times 10\sqrt{3}\times OM$

$\therefore OM=5cm$ Ans.