Question

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

Answer 1

If two circles touch internally, then distance between their centres is equal to the difference of their radii.

So, AB = (5 − 3) cm = 2 cm.

Also, the common chord PQ is the perpendicular bisector of AB.

Therefore, AC = CB = `1/2` AB = 1 cm

In right ΔACP, we have AP² = AC² + CP²

`=>` 5² = 1² + CP²

`=>` CP² = 25 – 1 = 24

`=> CP = sqrt(24)= 2 sqrt(6)  cm`

Now, PQ = 2 CP 

= `2 xx 2sqrt(6)  cm`

= `4sqrt(6)  cm`