Question

In the adjoining figure, $O$ is the centre of the circle and $B$ is a point of contact. Seg $OE\perp$ seg $AD$, $AB=12,AC=8$, find
i. $AD$
ii. $DC$
iii. DE.

Answer 1

i. Line $AB$ is the tangent at point $B$ and seg $AD$ is the secant. [Given]
$\therefore AC\times AD={AB}^2[$ Tangent secant segments theorem]
$$\begin{array}{cc}& \therefore 8\times AD=122\\ & \therefore 8\times AD=144\\ & \therefore AD=\frac{144}{8}\\ & \therefore AD=18\text{units}\end{array}$$
ii. $AD=AC+DC[A-C-D]$
$\therefore 18=8+DC$
$\therefore DC=18-8$
$\therefore DC=10$ units
iii. seg $OE\perp$ seg $AD\left[Given\right]$
i.e. seg $OE\perp$ seg $CD[A-C-D]$

$\therefore DE=\frac{1}{2}DC$ [Perpendicular drawn from the centre of the circle to the chord bisects the chord]
$$=\frac{1}{2}\times 10$$
$\therefore DE=5$ units