Question

In $△ABC$, $D$ and $E$ are midpoints of $AB$ and $AC$ respectively. Find the ratio of the areas of $△ADE$ and $△ABC$.

• A. $1:4$
• B. $2:5$
• C. $1:6$
• D. $3:7$

Answer 1

The correct option is A $1:4$

Given: $D$ and $E$ are mid points of $AB$ and $AC$
By mid point theorem, $DE\parallel BC$
In $△ADE$ and $△ABC$
$\angle DAE=\angle BAC$ (Common)
$\angle ADE=\angle ABC$ (Corresponding angles)
$\angle AED=\angle ACB$ (Corresponding angles)
Thus, $△ABC\sim △ADE$ ($AAA$ rule)
Hence, $\frac{Area\left(△ADE\right)}{Area\left(△ABC\right)}=\frac{A{D}^2}{A{B}^2}$ (Similar triangle property)
$\frac{Area\left(△ADE\right)}{Area\left(△ABC\right)}=\frac{A{D}^2}{(2AD{)}^2}$
$\frac{Area\left(△ADE\right)}{Area\left(△ABC\right)}=1:4$