Question

Two concentric circles are of radii 7 cm and r cm respectively, where r > 7. A chord of the larger
Circle, of length 48 cm, touches the smaller circle. Find the value of r.

Answer 1

Let O be the centre of the two concentric circles. Let PQ be the chord of larger circle touching the smaller circle at M. This can be represented diagrammatically as:

We have PQ = 48 cm.
Radius of the smaller circle, OM = 7 cm
Let the radius of the larger circle be r, i.e. OP = r
Since PQ is a tangent to the inner circle, OM $\Lambda$PQ
Thus, OM bisects PQ.
$\Rightarrow PM=MQ=\frac{48}{2}cm=24cm$
Now applying Pythagoras Theorem in $\Delta$OPM
$O{P}^2=O{M}^2+P{M}^2$
$\Rightarrow O{P}^2=(7cm{)}^2+(24cm{)}^2=(49+576)c{m}^2=625c{m}^2=(25cm{)}^2$
$\Rightarrow OP=25cm$
$\therefore$ Radius of the larger circle is 25 cm.
Thus, the value of r is 25 cm.