11. Construction of a pair of tangents to a circle from an external point

Draw a circle of diameter $6$ $cm$ from a point $P$, which is $8$ $cm$ away from its centre. Draw the two tangents $PA$ and $PB$ to the circle and measure their lengths.

 

Given:

 

The diameter of the circle $=$ $6$ $cm$.

Radius $=$ $\frac{6}{2}$

 

$=$ $3$ $cm$

 

Rough Sketch:

 

 

Construction:

 

Step $1$: With $O$ as the centre, draw a circle of radius $3$ $cm$.

 

Step $2$: Draw a line $OP$ of length $8$ $cm$.

 

Step $3$: Draw a perpendicular bisector of $OP$, which cuts $OP$ at $M$.

 

Step $4$: With $M$ as the centre and $MO$ as the radius, draw a circle that cuts the previous circle at $A$ and $B$.

 

Step $5$: Join $AP$ and $BP$. $AP$ and $BP$ are the required tangents. Thus the length of the
tangents are $PA = PB = 7.4$ $cm$.

 

Verification:

 

In the right angle triangle $OAP$ by the Pythagoras theorem, we have:

 

$OP^2$ $=$ $OA^2$ $+$ $PA^2$

 

$\Rightarrow$ $PA^2$ $=$ $OP^2$ $-$ $OA^2$

 

$PA^2$ $=$ $8^2$ $+$ $3^2$

 

$PA^2$ $=$ $64 – 9$

 

$PA^2$ $=$ $55$

 

$\Rightarrow PA = \sqrt{55}$

 

$PA$ $=$ $7.4$ $cm$ (approximately) .