Question

In the figure above, $P,Q$ and $R$ lie on the same line. $P$ is the center of the larger circle and $Q$ is the center of the smaller circle. If the radius of the larger circle is $4$, what is the radius of the smaller circle?

• A. $1$
• B. $2$
• C. $4$
• D. $8$
• E. $16$

Answer 1

The correct option is B $2$

Given that

$P$, $Q$, $R$ lie on the same line.

Radius of the larger circle is ${}^{\prime }{4}^{\prime }$.

Let ${}^{\prime }{r}^{\prime }$ be the radius of the smaller circle.

As $Q$ is the center of the smaller circle and $P$, $R$ are the points on the circumference of the smaller circle, lengths of $PQ$ and $QR$ are equal to the radius of the smaller circle.

Hence, $PQ$ $=$ $QR$ $=$ $r$

As $P$ is the center of the larger circle and $R$ is a point on the circumference of the larger circle, radius of the larger circle is ${}^{\prime }P{R}^{\prime }$.

Hence, $PR$ $=$ $4$.

As $P$, $Q$, $R$ lie on the same line,

$PR$ $=$ $PQ$ $+$ $QR$

$\Rightarrow 4$ $=$ $r$ $+$ $r$

$\Rightarrow 2r$ $=$ $4$

$\Rightarrow r$ $=$ $2$

Therefore, Radius of the smaller circle is ${}^{\prime }{2}^{\prime }$.