Question

The figure obtained by joining the mid-points of the adjacent sides of a rectangle of sides $8cm$ and $6cm$ is

• A. a rectangle of area $24{cm}^2$
• B. a square of area $25{cm}^2$
• C. a rhombus of area $24{cm}^2$
• D. a trapezium of area $12{cm}^2$

Answer 1

The correct option is C a rhombus of area $24{cm}^2$

We know that diagonals of a rectangle are congruent.

$\therefore AC=BD$
Join points $AC$ (diagonal)

Consider $\Delta ABC$
Since $P$ and $Q$ are midpoints of the two sides of the triangle,
we can say that $PQ||AC$ and $PQ=\frac{1}{2}AC$ ..(i)
The segment joining midpoints of any $2$ sides of a triangle is parallel to the $3^{rd}$ side and is half of it
Similarly in $\Delta ADC$, $RS\parallel AC$ and $RS=\frac{1}{2}AC$ ..(ii)

From (i) and (ii), $PQ=RS$
Similarly, if we join $BD$ (diagonal), we can prove that $PS=QR$
Since the diagonals of rectangle are equal in length, we can say $PQ=RS=PS=QR$
Thus, $PQRS$ is a rhombus
Area of rhombus $=\frac{1}{2}\times \text{product of diagonals}$

$=\frac{1}{2}\times QS\times PR$

$=\frac{1}{2}\times 8\times 6=24{cm}^2$

Hence, option C.