Question

From a point $P$ , the length of the tangent to a circle is $15cm$ and the distance of $P$ from the center of the circle is $17cm$ . Then what is the radius of the circle ?

Answer 1

Let the center of circle be $O$ so that $PO=17cm$
Tangent is $PB$ whose length is $15cm$
$OB$ is radius as shown
Now we know that radius is perpendicular to point of contact
Hence $OB$ is perpendicular to $PB$
Hence $\angle PBO={90}^{\circ }$
Consider $\Delta PBO$
Using Pythagoras theorem
$\Rightarrow P{B}^2+O{B}^2=P{O}^2$
$\Rightarrow {15}^2+O{B}^2={17}^2$
$\Rightarrow O{B}^2={17}^2-{15}^2$
$\Rightarrow O{B}^2=289-225$
$\Rightarrow O{B}^2=64$
$\Rightarrow OB=\pm 8$
As length cannot be negative
$\Rightarrow OB=8cm$
Hence length of radius is $8cm$