Tangents `PA` and `PB` are drawn from an external point `P` to two concentric circles with centre `O` and radii `8cm` and `5cm` respectively, as shown in the figure. If `AP=15cm` then find the length of `BP`.

Explanation

We have
`OAbotAP` and `OB bot BP` [`:'` the tangents at any point of a circle is perpendicular to the radius through the point of contact].
Join `OP`.
In right `DeltaOAP`, we have
`OA=8cm`, `AP=15cm`.
`:.OP^(2)=OA^(2)+AP^(2)`
[by Pythogoras's theorem]
`impliesOP=sqrt(OA^(2)+AP^(2))=sqrt(8^(2)+15^(2))cm=sqrt(289)cm=17cm`.
In right `DeltaOBP`, we have
`OB=5cm`, `OP=17cm`
`:.OP^(2)=OB^(2)+BP^(2)`[ by Pythagoras' theorem]
`impliesBP=sqrt(OP^(2)-OB^(2))=sqrt(17^(2)-5^(2))cm=sqrt(264)cm`
Thus, the length of `BP=sqrt(264)cm=16.25cm` (approx).