Question

P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC ⊥ BD. Prove that PQRS is a square.

Answer 1

According to the question,

We have,

P is the mid-point of the sides AB

Q is the mid-point of the sides BC

R is the mid-point of the sides CD

S is the mid-point of the sides DA

Also,

AC ⊥ BD

And AC = BD

In ΔADC, by mid-point theorem,

SR = ½ AC

And, SR||AC

In ΔABC, by mid-point theorem,

PQ = ½ AC

And, PQ||AC

So, we have,

PO||SR and PQ = SR = ½ AC

Now, in ΔABD, by mid-point theorem,

SP||BD and SP = ½ BD = ½ AC

In ΔBCD, by mid-point theorem,

RQ||BD and RQ = ½ BD = ½ AC

SP = RQ = ½ AC

PQ = SR = SP = RQ

Thus, we get that,

All four sides are equal.

Considering the quadrilateral EOFR,

OE||FR, OF||ER

∠EOF = ∠ERF = 90^o (Opposite angles of parallelogram)

∠QRS = 90^o

Hence, PQRS is a square.