Question

Construct the circumcircle and incircle of an equilateral $\Delta$ XYZ with side $6.5$cm and centre O. Find the ratio of the radii of incircle and circumcircle.

Answer 1

Circumcenter can be found using the perpendicular bisectors, and incenter using angle bisectors.

Now taking $R$ as circumradius and $r$ as inradius, we can draw the circumcircle and incircle.

In triangle $OXZ$ , $\angle XOZ={120}^0$ and $\angle OXZ=\angle OZX={30}^0$

Measuring $r$ and $R$ using ruler, we can get $\frac{r}{R}=\frac{1}{2}$

$\text{Alternate Method:}$

In triangle $OXZ$ , $\angle XOZ={120}^0$ and $\angle OXZ=\angle OZX={30}^0$

Therefore by sine rule we get $R=6.5\times \frac{\sin {30}^0}{\sin {120}^0}=\frac{6.5}{\sqrt{3}}$

By sine rule, we get $\frac{r}{\sin {30}^0}=\frac{R}{\sin {90}^0}$

$\Rightarrow r=R\sin {30}^0=\frac{R}{2}$

$\Rightarrow \frac{r}{R}=\frac{1}{2}$