Question

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Answer 1

Let the two concentric circles be centered at point $O$. And let $PQ$ be the chord of the larger circle which touches the smaller circle at point $A$. Therefore, $PQ$ is tangent to the smaller circle.
$OA⊥PQ$ (As $OA$ is the radius of the circle)
Applying Pythagoras theorem in $ΔOAP$, we obtain
$OA ^2 + AP ^2 = OP^2$
$3 ^2 + AP ^2 = 5 ^2$
$9 + AP ^2 = 25$
$AP ^2 = 16$
$AP = 4$
In $ΔOPQ$,
Since $OA ⊥ PQ$
$AP = AQ$ (Perpendicular from the center of the circle bisects the chord)
∴ $PQ = 2AP = 2 × 4 = 8$

Therefore, the length of the chord of the larger circle is $8$ cm.