Question

The line segments joining the mid points of a quadrilateral form a __________.

• A. parallelogram
• B. kite
• C. rhombus
• D. triangle

Answer 1

The correct option is A parallelogram


Consider a quadrilateral ABCD.
P, Q, R, and S are the mid-points of sides AB, BC, CD and DA respectively.

Construction: Join AC.

The mid-point theorem states that a line segment joining mid points of two sides of a triangle is parallel to the third side and is half of it.

Consider the triangle DAC.
By mid point theorem,
$SR\parallel AC\text{and}SR=\frac{1}{2}AC...\left(i\right)$

Similarly, in triangle BAC, by mid point theorem,
$PQ\parallel AC\text{and}PQ=\frac{1}{2}AC...\left(ii\right)$

$\text{From (i) and (ii),}⟹SR\parallel AC\parallel PQ...\left(iii\right)⟹SR=PQ=\frac{1}{2}AC...\left(iv\right)$

$\text{From (iii) and (iv),}SP\parallel RQSP=RQ$

Opposite sides of the quadrilateral PQRS are parallel and equal.
Hence, PQRS is a parallelogram.