Question

In the following figure, point $A$ is the centre of the circle, seg $AM$ $\perp$ line $MN,AN=10cm$, line $MN$ is tangent at $M$. Determine radius of the circle if $MN=5cm.$

Answer 1

$\angle AMN={90}^o$ [$\because$ Tangent to the radius at the point of contact.]
In right angled $\Delta AMN$
By Pythagoras theorem
$A{M}^2+M{N}^2=A{N}^2$
$A{M}^2=A{N}^2-M{N}^2$
$=(10{)}^2-(5{)}^2$
$=100-25$
$=75$
$\therefore AM=\sqrt{75}$
$\therefore AM=5\sqrt{3}$ cm.
$\therefore$ The radius of the circle is $5\sqrt{3}$ cm.