ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

Given: AD = BC

To prove: PQRS is a rhombus

Theorem Used:

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

Proof:

In ΔBAD,

P and S are the mid points of sides AB and BD,

So, by midpoint theorem,

PS||AD and PS = 1/2 AD … (i)

In ΔCAD,

R and Q are the mid points of CD and AC,

So, by midpoint theorem

QR||AD and QR = 1/2 AD … (ii)

Compare (i) and (ii)

PS||QR and PS = QR

Since one pair of opposite sides is equal as well as parallel then

PQRS is a parallelogram … (iii)

Now, In ΔABC, by midpoint theorem

PQ||BC and PQ = 1/2 BC … (iv)

And Ad = BC … (v)

Compare equations (i) (iv) and (v)

PS = PQ … (vi)

From (iii) and (vi)

Since PQRS is a parallelogram with PS = PQ then PQRS is a rhombus.