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In figure below, if $ \mathrm{DE} \| \mathrm{BC} $, find the ratio of ar (ADE) and ar (DECB).
"
Given:
\( \mathrm{DE} \| \mathrm{BC} \)
To do:
We have to find the ratio of ar (ADE) and ar (DECB).
Solution:
In $\triangle A B C$ and $\triangle A D E$,
$\angle A B C=\angle A D E$ (Corresponding angles)
$\angle A C B=\angle A E D$ (Corresponding angles)
$\angle A =\angle A$
Therefore, by AA similarity,
$\triangle A B C \sim \triangle A E D$
This implies,
$\frac{\operatorname{ar}(\triangle A D E)}{\operatorname{ar}(\triangle A B C)}=(\frac{DE}{BC})^2$
$=\frac{(6)^{2}}{(12)^{2}}$
$=(\frac{1}{2})^{2}$
$=\frac{1}{4}$
Let $ar (\triangle A D E)=k$
This implies,
$ar (\triangle A B C)=4 k$
$ar (D E C B)=\operatorname{ar}(A B C)-\operatorname{ar}(A D E)$
$=4 k-k$
$=3 k$
Therefore,
$\operatorname{ar}(A D E): \operatorname{ar}(D E C B)=k: 3 k$
$=1: 3$
The ratio of ar (ADE) and ar (DECB) is $1:3$.