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

If $AD\ =\ 2\ cm$, $AB\ =\ 6\ cm$ and $AC\ =\ 9\ cm$, find $AE$.

"

Given:

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

$AD\ =\ 2\ cm$, $AB\ =\ 6\ cm$ and $AC\ =\ 9\ cm$.

To do:

We have to find the measure of $AE$.

Solution:

$DE\ ||\ BC$ (given)

Therefore,

By Basic proportionality theorem,

$\frac{AD}{DB}\ =\ \frac{AE}{EC}$

This implies,

$\frac{DB}{AD}\ =\ \frac{EC}{AE}$

Adding $1$ on both sides,

$\frac{DB}{AD} + 1 = \frac{EC}{AE} + 1$

$\frac{DB+AD}{AD}=\frac{EC+AE}{AE}$

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

$\frac{6}{2}=\frac{9}{AE}$

$AE=\frac{9}{3}$

$AE=3 cm$

The measure of $AE$ is $3 cm$.

- Related Articles
- In a $Δ$ ABC, D and E are points on the sides AB and AC respectively such that DE $||$ BC.If AD $=$ 6 cm, DB $=$ 9 cm and AE $=$ 8 cm, find AC."\n
- In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $AD\ =\ 4\ cm$, $DB\ =\ 4.5\ cm$ and $AE\ =\ 8\ cm$, find $AC$. "\n
- 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 $CE$. "\n
- In a $Δ\ ABC,$ $D$ and $E$ are points on the sides $AB$ and $AC$ respectively. For each of the following cases show that $DE\ ∥\ BC$: $AB\ =\ 12\ cm$, $AD\ =\ 8\ cm$, $AE\ =\ 12\ cm$, and $AC\ =\ 18\ cm$. "\n
- In a $Δ\ ABC,$ $D$ and $E$ are points on the sides $AB$ and $AC$ respectively. For each of the following cases show that $DE\ ∥\ BC$: $AB\ =\ 5.6\ cm$, $AD\ =\ 1.4\ cm$, $AC\ =\ 7.2\ cm$, and $AE\ =\ 1.8\ cm$. "\n
- In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $\frac{AD}{DB}\ =\ \frac{2}{3}$ and $AC\ =\ 18\ cm$, find $AE$. "\n
- In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $\frac{AD}{DB}\ =\ \frac{3}{4}$ and $AC\ =\ 15\ cm$, find $AE$. "\n
- $D$ and $E$ are the points on the sides $AB$ and $AC$ respectively of a $\triangle ABC$ such that: $AD = 8\ cm, DB = 12\ cm, AE = 6\ cm$ and $CE = 9\ cm$. Prove that $BC = \frac{5}{2}DE$.
- In a $Δ\ ABC,$ $D$ and $E$ are points on the sides $AB$ and $AC$ respectively. For each of the following cases show that $DE\ ∥\ BC$: $AB\ =\ 10.8\ cm$, $BD\ =\ 4.5\ cm$, $AC\ =\ 4.8\ cm$, and $AE\ =\ 2.8\ cm$. "\n
- In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $AD\ =\ x\ cm$, $DB\ =\ x\ –\ 2\ cm$, $AE\ =\ x\ +\ 2\ cm$, and $EC\ =\ x\ –\ 1\ cm$, find the value of $x$. "\n
- In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $AD\ =\ 4\ cm$, $AE\ =\ 8\ cm$, $DB\ =\ x\ –\ 4\ cm$ and $EC\ =\ 3x\ –\ 19$, find $x$. "\n
- In a $Δ\ ABC,$ $D$ and $E$ are points on the sides $AB$ and $AC$ respectively. For each of the following cases show that $DE\ ∥\ BC$: $AD\ =\ 5.7\ cm$, $BD\ =\ 9.5\ cm$, $AE\ =\ 3.3\ cm$, and $EC\ =\ 5.5\ cm$. "\n
- In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $\frac{AD}{BD}\ =\ \frac{4}{5}$ and $EC\ =\ 2.5\ cm$, find $AE$. "\n
- In a $Δ\ ABC$, $P$ and $Q$ are the points on sides $AB$ and $AC$ respectively, such that $PQ\ ∥\ BC$. If $AP\ =\ 2.4\ cm$, $AQ\ =\ 2\ cm$, $QC\ =\ 3\ cm$ and $BC\ =\ 6\ cm$. Find $AB$ and $PQ$. "\n
- In a triangle ABC, DE is parallel to BC. If AB = 7.2 cm; AC = 9 cm; and AD = 1.8 cm; Find AE."\n

##### Kickstart Your Career

Get certified by completing the course

Get Started