Question

In the given figure, if $\angle ACB=\angle CDA,AC=8cm$ and $AD=3cm$, then find $BD$.

Answer 1

Given:

$AC=8cm,AD=3cm$ and $\angle ACB=\angle CDA$

From the figure,

In $\Delta ADC$ and $\Delta ACB$

$\angle A=\angle A$ [Since, common angle]

$(\angle ACB=\angle ADC$ [Since, given]

So, by A.A. Similarity criterion,

$\Delta ADC\sim \Delta ACB$

$\Rightarrow$ Corresponding sides are in the same ratio.

$\Rightarrow \frac{AC}{AD}=\frac{AB}{AC}$

$\Rightarrow \frac{8cm}{3cm}=\frac{AB}{8cm}$

$\Rightarrow \frac{8}{3}\times 8cm=AB$

$\Rightarrow AB=\frac{64}{3}cm$ ---(1)

Now, from the figure,

$BD=AB-AD$

$\Rightarrow BD=\frac{64}{3}cm-3cm$

$\Rightarrow BD=\frac{64-9}{3}cm$

$\Rightarrow BD=\frac{55}{3}cm$

$\therefore BD=18\frac{1}{3}cm$