Question

In the figure given, $O$ is the centre of the circle. $AB$ and $CD$ are two chords of the circle. $OM$ is perpendicular to $CD$ and $ON$ is perpendicular to $AB$. $AB=24cm,ON=5cm,OM=12cm$. Find the length of chord $CD$

Answer 1

Given: $AB=24cm;ON=5cm,OM=12cm$.
$\because ON\perp AB$
$N$ is the mid point of $AB$.
$\therefore AN=\frac{24}{2}cm=12cm$.
Now from $△ANO,$

$A{O}^2=O{N}^2+A{N}^2$
$\Rightarrow r^2=5^2+{12}^2$ $(\because AO=CO=r)$
$\Rightarrow r^2=25+144$

$\Rightarrow r=13$

So,$AO=CO=13cm$

From $\Delta CMO$,

$C{M}^2=C{O}^2-O{M}^2$

$\Rightarrow C{M}^2={13}^2-{12}^2$

$\Rightarrow C{M}^2=169-144$
$\Rightarrow C{M}^2=25$
$\Rightarrow CM=5$
As $OM\perp CD,M$ is the mid point of $CD$.
$\therefore CD=2CM=2\times 5cm=10cm$.