Question

A quadrilateral formed by using the mid-points of the sides of any quadrilateral is always a

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

Answer 1

The correct option is D Parallelogram


Consider a quadrilateral ABCD such that mid-points on its sides AB, BC, CD and DA are P, Q, R and S respectively.
In $\Delta$ADC, S and R are the mid-points of sides AD and CD respectively.
In a triangle, the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.
$\therefore SR||AC$ and $SR=\frac{1}{2}AC$.....(i)
In $\Delta ABC$, P and Q are mid-points of sides AB and BC respectively. Therefore, by using mid-point theorem,
$PQ||AC$ and $PQ=\frac{1}{2}AC$ .....(ii)
Using (i) and (ii), we obtain
$PQ||SR$ and $PQ=SR$ …..(iii)
Clearly, one pair of opposite sides of quadrilateral PQRS is parallel and equal.
Hence, PQRS is a parallelogram.
Hence, the correct answer is option (d).