Question

A circle is inscribed in an equilateral $△ABC$. If the side of the triangle is 21 cm, what is the radius of the circle?

• A. $\frac{21}{2\sqrt{3}}cm$
• B. $\frac{21}{3\sqrt{2}}cm$
• C. $\frac{21}{2\times 3}cm$
• D. $\frac{21}{\sqrt{2}\times \sqrt{3}}cm$

Answer 1

The correct option is A $\frac{21}{2\sqrt{3}}cm$

Given: $ABC$ is equilateral triangle with $side=21cm$

$O$ is incentre of triangle . $o$ is centroid also because $ABC$ is equilateral triangle.

in $△ADC$

$\angle D=90°$

$\angle DAC=30°$

$\angle ACD=60°$

So, by $30-60-90$ triangle property, $AC=2x,DC=x,AD=\sqrt{3}x$

$2x=21⟹x=\frac{21}{2}$

$AD=\frac{21\sqrt{3}}{2}$

$O$ divides $AD$ in $2:1$ ratio

$\therefore OD=\frac{21\sqrt{3}}{2\times 3}=\frac{7\sqrt{3}}{2}=\frac{21}{2\sqrt{3}}$