In a $Δ\ ABC$, $D$ and $E$ are points on $AB$ and $AC$ respectively, such that $DE\ ∥\ BC$. If $AD\ =\ 2.4\ cm$, $AE\ =\ 3.2\ cm$, $DE\ =\ 2\ cm$ and $BC\ =\ 5\ cm$. Find $BD$ and $CE$.
"

Given:

In a $Δ\ ABC$, $D$ and $E$ are points on $AB$ and $AC$ respectively, such that $DE\ ∥\ BC$.

$AD\ =\ 2.4\ cm$, $AE\ =\ 3.2\ cm$, $DE\ =\ 2\ cm$ and $BC\ =\ 5\ cm$.

To do:

Here, we have to find $BD$ and $CE$.

Solution:

$DE || BC$, $AB$ is transversal.

$\angle APQ = \angle ABC$ (corresponding angles)

$DE || BC$, $AC$ is transversal.

$\angle AED = \angle ACB$ (corresponding angles)

In  $\vartriangle ADE$ and $\vartriangle  ABC$,

$\angle ADE=\angle ABC$

$\angle AED=\angle ACB$

$\vartriangle ADE = \vartriangle ABC$ (By AA similarity criteria)

$\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$  (CPCT)

$\frac{AD}{AB} = \frac{DE}{BC}$

$\frac{2.4} {(2.4 + DB)} = \frac{2}{5}$  ($AB = AD + DB$)

$2.4 + DB = \frac{2.4\times5}{2}$

$2.4 + DB = 6$

$DB = 6 – 2.4$

$DB = 3.6 cm$

Similarly,

$\frac{AE}{AC} = \frac{DE}{BC}$

$\frac{3.2}{(3.2 + EC)} = \frac{2}{5}$    ($AC = AE + EC$)

$3.2 + EC = \frac{3.2\times5}{2}$

$3.2 + EC = 8$

$EC = 8 – 3.2$

$EC = 4.8 cm$

Therefore,

$BD = 3.6 cm$ and $CE = 4.8 cm$.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

32 Views

Kickstart Your Career

Get certified by completing the course

Get Started