Question

An equilateral triangle ABC of side 4 cm is inscribed in a circle which is centered at O, as shown below. Then the radius of this circle is

• A. $\frac{5}{\sqrt{3}}$ cm
• B. $\frac{4}{\sqrt{3}}$ cm
• C. $\frac{2}{\sqrt{3}}$ cm
• D. $\frac{1}{\sqrt{3}}$ cm

Answer 1

The correct option is B

$\frac{4}{\sqrt{3}}$ cm

Given, $△$ABC is an equilateral triangle each side of which is 4 cm and is inscribed in a circle which is centered at O.

Let OE be the radius($r$) of this circle which is perpendicular to BC and cuts BC at point D.

We know that the perpendicular from the centre of the circle to chord bisects the chord.

So, we must have, BD = 2 cm ($\because$ Given, BC = 4 cm)

Also, we know that a side of an equilateral triangle drawn with vertices on a circle bisects the radius perpendicular to it.

So, OD = $\frac{r}{2}$ cm

By applying Pythagoras theorem to $△$BOD, we get,

$B{O}^2=O{D}^2+B{D}^2$

$⟹r^2=(\frac{r}{2}{)}^2+2^2$

$⟹r^2-\frac{r^2}{4}=4$

$⟹\frac{3r^2}{4}=4$

$⟹r^2=\frac{4\times 4}{3}$

$⟹r=\sqrt{\frac{4\times 4}{3}}$

$\therefore r=\frac{4}{\sqrt{3}}$ cm