Question

Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of $\angle APB$ is
A. \[{30^ \circ }\]
B. \[{45^ \circ }\]
C. \[{60^ \circ }\]
D. ${90^ \circ }$

Answer 1

Hint: In these types of questions construct a rough diagram with the help of given information and use the angle sum property of triangle i.e. Sum of all angles =${180^ \circ }$ and find the value of $\angle $APB.

Complete step-by-step answer:

First let’s draw a rough diagram with the help of the given information.

Let’s draw two circles touching each other externally at point P.

Draw a common tangent to both the circles.

Then construct line PB, PA and PC touching tangent where PC divides the tangent into AC and BC.

Let, $\angle $PAB = $\alpha $ and $\angle $PBA = $\beta $

In triangle ACP,

AC = CP by the theorem of tangent to a circle i.e. when two tangents are drawn from an external point to a circle the length of the tangents are equal to each other.

Since AC = CP therefore the $\angle $PAB =$\angle $APC

So, $\angle $APC = $\alpha $

In triangle BPC,

PC = BC by the theorem of tangent to a circle i.e. when two tangents are drawn from an external point to a circle the length of the tangents are equal to each other.

Since AC = CP therefore the $\angle $PBA =$\angle $BPC

So, $\angle $BPC = $\beta $

In triangle APB

$\angle $PAB +$\angle $PBA +$\angle $APB = 180 (by the angle sum property of triangle)

$\alpha $ + $\beta $ + ($\alpha $ +$\beta $) = ${180^ \circ }$

$\alpha $+ $\beta $ +$\alpha $+ $\beta $ =${180^ \circ }$

2 ($\alpha $ +$\beta $) = ${180^ \circ }$

($\alpha $+$\beta $) = ${90^ \circ }$

So, $\angle $APB = ${90^ \circ }$

Note: In these types of questions first draw a rough diagram with the help of given information then let $\angle $PAB = $\alpha $ and $\angle $PBA = $\beta $ then use them to show the $\angle $PAB =$\angle $APC and $\angle $PBA =$\angle $BPC with the help of tangent to a circle theorem and then use angle sum property of triangle i.e.

Sum of all angles =${180^ \circ }$ and find the value of $\angle $APB.