# Find the radius of a circle inscribed in a pendant

The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector OAB of a circle centre O, of radius 6 cm, and angle $AOB = \frac{\pi}{3}$. The circle C, inside the sector, touches the two straight edges, OA and OB, and the arc AB as shown.

Find the radius of the circle C.

I have no clue how to approach this. Help please.

• A good general rule in any geometry problem where you have circles tangent to things is to draw all the radii of the circles to the points of tangency. Apr 19, 2018 at 23:02

still no clue?

You must use the commonly abused property of a tangent line in a circle. Any tangent line touching in the circle can produce a perpendicular line. Projecting the radius of C into the side of the sector. See the picture. You can get:

$$\sin \frac{\eta}{2} = \frac{r}{R-r}$$

Let $P$ be the center of circle $C$ and $Q$ be a point on $AO$ such that $CQ\perp AO$.

It can be shown that $OP$ is bisecting $\angle AOB$. Because of this, $\triangle OCQ$ has angles measuring $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$.

Let $r$ be the radius of the circle. From this: $CQ=r$, $CO=2r$, and $QO=r\sqrt{3}$.

From here, try finding the value of $r$ with what you know.