Question

The radii of two concentric circles are $17cm$ and $10cm$. A line segment $PQRS$ cuts the larger circle at $Q$ and $R$. If $QR=12cm$, find the length $PQ$.

Answer 1

Given:$OP=17$cm and $OR=OQ=10$cm

$QR=12$cm

To find $PQ$

Drop $OM$ perpendicular to $QR$

Since perpendicular from center to chord bisects the chord.

$MQ=6$cm

In $△OMQ,MQ=6$cm,$OQ=10$cm

Using Pythagoras theorem,

$OM=\sqrt{100-36}=\sqrt{64}=8$cm

In $△OMP,OM=8$cm,$OP=17$cm

Using Pythagoras theorem,

$PM=\sqrt{289-64}=\sqrt{225}=15$cm

Clearly, $PQ=PM-QN=15-6=9$cm