Question
In the given figure, if $\angle\text{ADE}=\angle\text{B},$ show that $\triangle\text{ADE}\sim\triangle\text{ABC}.$ If AD = 3.8cm, AE = 3.6cm, BE = 2.1cm and BC = 4.2cm, find DE.
✓
Answer
Given: $\angle\text{ADE}=\angle\text{B},$ AD = 3.8cm, AE = 3.6cm, BE = 2.1cm, BC = 4.2cm
Proof:
In $\triangle\text{ADE}$ and $\triangle\text{ABC},$
$\angle\text{A}=\angle\text{A}$ (common)
$\angle\text{ADE}=\angle\text{B}$ (given)
Therefore, $\triangle\text{ADE}\sim\triangle\text{ABC}$ (AA Criterion)
$\Rightarrow\frac{\text{AD}}{\text{AB}}=\frac{\text{DE}}{\text{BC}}$
$\Rightarrow\frac{3.8}{(3.6+2.1)}=\frac{\text{x}}{4.2 }(\text{DE}=\text{x})$
$\Rightarrow\frac{3.8}{5.7}=\frac{\text{x}}{4.2}$
$\text{x}=\frac{3.8\times4.2}{5.7}=2.8\text{cm}$
Hence, DE = 2.8cm
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