Question

Question 8
In figure, if DEIIBC, find the ratio of ar $\left(\Delta ADE\right)$ and ar $\left(DECB\right)$.

Answer 1

Given, DE || BC, DE= 6 cm and BC = 12 cm
$In\Delta ABCand\Delta ADE$,

$\angle ABC=\angle ADE\left[correspondingangles\right]$
$\angle ACB=\angle AED\left[correspondingangles\right]$
and $\angle A=\angle A\left[commonangle\right]$
$\therefore \Delta ABC\sim \Delta ADE\left[byAAAsimilaritycriterion\right]$

$Then,\frac{ar\left(\Delta ADE\right)}{ar\left(\Delta ABC\right)}=\frac{(DE{)}^2}{(BC{)}^2}$
(By similar triangle property)

$=\frac{(6{)}^2}{(12{)}^2}={\left(\frac{1}{2}\right)}^2$

$\Rightarrow$ $\frac{ar\left(\Delta ADE\right)}{ar\Delta ABC}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$

Let ar $\left(\Delta ADE\right)=k$, then ar $\left(\Delta ABC\right)=4k$

Now, $ar\left(DECB\right)=ar\left(ABC\right)-ar\left(ADE\right)=4k-k=3k$
$\therefore$ Required ratio = ar(ADE) : ar(DECB) = k : 3k = 1 : 3