Question

The quadrilateral formed by joining the mid-points of the sides of a square is always a

• A. Rectangle
• B. Square
• C. Kite
• D. Rhombus

Answer 1

The correct option is B Square

Consider a square ABCD and let P, Q, R, S be the mid-points of AB, BC, CD and AD respectively.

$\therefore AP=PB=BQ=CQ=CR=RD=DS=SA$
$=\frac{1}{2}AB=\frac{1}{2}BC=\frac{1}{2}CD=\frac{1}{2}AD=K$
In ∆APS, by Pythagoras theorem,
$(PS{)}^2=(AP{)}^2+(AS{)}^2$
$\Rightarrow (PS{)}^2=K^2+K^2$
$\Rightarrow PS=\sqrt{2}K$
Similarly, $PQ=QR=RS=PS=\sqrt{2}K$ ........(i)
Now in ∆APS, by Angle sum property,
$\angle ASP+\angle APS+\angle SAP=180°$
$\Rightarrow \angle ASP+\angle ASP=180°–90°=90°$ $(\because AS=AP)$
$\Rightarrow \angle ASP=45°$
Similarly, $\angle DSR=45°$
Now, DSA is a straight line.
$\therefore \angle DSR+\angle RSP+\angle PSA=180°$
$\Rightarrow 45°+\angle RSP+45°=180°$
$\Rightarrow \angle RSP=90°$
Similarly, $\angle SPQ=\angle PQR=\angle QRS=\angle RSP=90°$ …..(ii)
From (i) and (ii), it can be concluded that PQRS is also a square.
Hence, the correct answer is option (b).