Question

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

Answer 1

Hint: We will first join O and P. Then after using Pythagoras theorem which is in a right-angled triangle \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\], we will find the length of OP. We will then again apply the Pythagoras theorem, to find the length of BP.
We know that PA and PB are tangents from point P to two concentric circles. This implies that they form right angles with the center O.

Complete step-by-step solution:
Let us join O and P.
Then our figure looks as,

In \[\vartriangle OAP\] , and\[\vartriangle OBP\],
\[\angle OAP={{90}^{\circ }}\] And \[\angle OBP={{90}^{\circ }}\]
\[\Rightarrow \]OP is the hypotenuse if these right angled triangle
We know that in a right angled triangle, by Pythagoras’s theorem,
 \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\]
Where c is the hypotenuse and ‘a’ and ‘b’ are the perpendicular sides of the triangle.
For\[\vartriangle OAP\], radius OA=8cm
By Pythagoras’s theorem,
  $ O{{P}^{2}}=O{{A}^{2}}+A{{P}^{2}} $
 $\Rightarrow O{{P}^{2}}={{8}^{2}}+{{(15)}^{2}}$
 $=64+225=289={{(17)}^{2}} $
 $\Rightarrow OP=17cm $ -------(1)
For\[\vartriangle OBP\], radius OB= 5cm
By Pythagoras’s theorem,
 $ O{{P}^{2}}=O{{B}^{2}}+B{{P}^{2}} $
 $ \Rightarrow B{{P}^{2}}=O{{P}^{2}}-O{{B}^{2}}$
 $ ={{(17)}^{2}}-{{(5)}^{2}} $
 $ =289-25$
 $ B{{P}^{2}}=264 $
 $ \Rightarrow BP=16.25cm $ ------------- (from equation (1) OP=17cm)
Hence, the length of side BP=16.25cm.

Note: It is known that the tangents of any point of a circle are perpendicular to the radius through the point of continent. In this question, tangents PA and PB touch the concentric circles at A and B. Hence, they form right angles with the centre O. While solving such questions, we should know when and how to use the properties of a circle. We should also remember that distance is a non-negative quantity. Hence, while taking square roots to find OP and BP, we will always consider positive square roots. While applying Pythagoras’s theorem, we should always first find the right angle and hypotenuse, as it can lead to error when we change position of hypotenuse with any other side.