Question

Two circles of radii 5 cm and 3 cm and centres A and B touch internally. If the perpendicular bisector of segment AB meets the bigger circle in P and Q, find the length of PQ.

Answer 1

A and B are the centers of the circles with radii 5cm and 3cm respectively. O is the mid-point of AB.

$AB=5-3=2cm.$

$AO=\frac{AB}{2}=1cm$ ....... ($\because$ O is the midpoint of AB)

Now in right angled $\Delta AOP,AO=1cm,AP=5cm$

By Pythagoras theorem,

$A{P}^2=A{O}^2+O{P}^2$

$\Rightarrow OP=\sqrt{A{P}^2-A{O}^2}=\sqrt{(5cm{)}^2–(1cm{)}^2}=\sqrt{24}$

Then, $PQ=2\times OP=2\times \sqrt{24}=4\sqrt{6}$