# In figure below, $DE\ âˆ¥\ BC$ such that $AE\ =\ (\frac{1}{4})AC$. If $AB\ =\ 6\ cm$, find $AD$.

"

**Given:**

**
**

In the given figure, $DE\ âˆ¥\ BC$ such that $AE\ =\ (\frac{1}{4})AC$ and $AB\ =\ 6\ cm$.

**
**

**To do:**

**
**

We have to find $AD$.

**Solution:**

**
**

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

$\angle A = \angle A$ (Common)

$\angle ADE = \angle ABC$ ($AB||QR$, Corresponding angles)

Therefore,

$\vartriangle ADE ∼ \vartriangle ABC$ (By AA similarity)

$\frac{AD}{AB} = \frac{AE}{AC}$ (Corresponding parts of similar triangles are proportional)

$\frac{AD}{6} = \frac{1}{4}$ ($AE\ =\ (\frac{1}{4})AC$, this implies, $\frac{AE}{AC} =\frac{1}{4}$)

$AD = \frac{6}{4}$

$AD = 1.5\ cm$

**The measure of $AD$ is $1.5\ cm$.**

- Related Articles
- 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
- $ABC$ is a triangle. $D$ is a point on $AB$ such that $AD = \frac{1}{4}AB$ and $E$ is a point on $AC$ such that $AE = \frac{1}{4}AC$. Prove that $DE =\frac{1}{4}BC$.
- 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 $AE$. "\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
- 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 triangle ABC, DE is parallel to BC. If AB = 7.2 cm; AC = 9 cm; and AD = 1.8 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 $\frac{AD}{DB}\ =\ \frac{2}{3}$ 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 $\frac{AD}{BD}\ =\ \frac{4}{5}$ and $EC\ =\ 2.5\ cm$, find $AE$. "\n
- In a $Δ$ ABC, D and E are points on the sides AB and AC respectively such that DE $||$ BC.If AD $=$ 6 cm, DB $=$ 9 cm and AE $=$ 8 cm, find AC."\n
- In figure below, $DE\ ||\ BC$. If $DE\ =\ 4\ m$, $BC\ =\ 6\ cm$ and $Area\ (ΔADE)\ =\ 16\ cm^2$, find the $Area\ of\ ΔABC$. "\n
- Construct a triangle $ABC$ such that $BC = 6\ cm, AB = 6\ cm$ and median $AD = 4\ cm$.
- 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 $AD\ =\ 4\ cm$, $AE\ =\ 8\ cm$, $DB\ =\ x\ –\ 4\ cm$ and $EC\ =\ 3x\ –\ 19$, find $x$. "\n
- In figure below, $AE$ is the bisector of the exterior $∠\ CAD$ meeting $BC$ produced in $E$. If $AB\ =\ 10\ cm$, $AC\ =\ 6\ cm$, and $BC\ =\ 12\ cm$, find $CE$.p>"\n

##### Kickstart Your Career

Get certified by completing the course

Get Started