Question

In $\Delta ABC$, point P,Q and R are the mid points of the sides AB, BC and CA respectively. If area of $\Delta ABC$ is 32 sq units, then area of $\Delta PQR$ is

• A. $8$ sq cm
• B. $16$ sq cm
• C. $64$ sq cm
• D. $24$ sq cm

Answer 1

The correct option is A $8$ sq cm

The line joining the midpoints of the sides of the triangle form four triangles, each of which is similar to the original triangle.

$\Delta ABC\sim \Delta PQR$

In $\Delta ABC$, $P$ and $R$ are mid points of $AB$ and $AC$ respectively.

$\therefore PR||BC$ (midpoint theorem)

In $\Delta ABC$ and $\Delta APR$:

$\angle A$ is common and $\angle APR=\angle ABC$ (corresponding angles)

Therefore, $\Delta ABC\sim \Delta APR$ (AA similarity)

In $\Delta ABC$ and $\Delta PQR$, since $P,Q,R$ are the midpoints of $AB,BC$ and $AC$ respectively,

$PR=\frac{1}{2}BC$; (midpoint theorem)

$\therefore \Delta ABC\sim \Delta PQR$ (SSS similarity)

$\frac{Ar\left(\Delta PQR\right)}{Ar\left(\Delta ABC\right)}=\frac{P{R}^2}{B{C}^2}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$

Now, it is given that area of $\Delta ABC$ is $32$ sq. units. Therefore, we have:

$\frac{Ar\left(\Delta PQR\right)}{32}=\frac{1}{4}Ar\left(\Delta PQR\right)=\frac{32}{4}Ar\left(\Delta PQR\right)=8$

Hence, the area of $\Delta PQR$ is $8$ sq units.