Question
In a $\triangle\text{ABC,D}\ \text{and E}$ are points on the sides AB and AC respectively such that DE || BC.
If AD = 2cm, AB = 6cm and AC = 9cm, find AE.
If AD = 2cm, AB = 6cm and AC = 9cm, find AE.
✓
Answer
We have,AD = 2cm, AB = 6cm
$\therefore$ DB = AB - AD
= 6 - 2
$\Rightarrow$ DB = 4cm
And, DE || BC
Therefore, by basic proportionality theorem, we have,
$\frac{\text{AD}}{\text{DB}}=\frac{\text{AE}}{\text{EC}}$
Taking reciprocal on both sides, we get,
$\frac{\text{DB}}{\text{AD}}=\frac{\text{EC}}{\text{AE}}$
$\frac{4}{2}=\frac{\text{EC}}{\text{AE}}$
Adding 1 on both sides, we get
$\frac{4}{2}+1=\frac{\text{EC}}{\text{AE}}+1$
$\Rightarrow\frac{4+2}{2}=\frac{\text{EC}+\text{AE}}{\text{AE}}$
$\Rightarrow\frac{6}{2}=\frac{\text{AC}}{\text{AE}}\ \ [\because\text{EC}+\text{AE}=\text{AC}]$
$\Rightarrow\frac{6}{2}=\frac{9}{\text{AE}}\ \ [\because\text{AC}=9\text{cm}]$
$\text{AE}=\frac{9\times2}{6}$
$\Rightarrow\text{AE}=3\text{cm}$
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