Question

The midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD are joined to form a quadrilateral. If AC = BD and AC ⊥ BD then prove that the quadrilateral formed is a square.

Answer 1

Consider △ ABC

We know that P and Q are the midpoints of AB and BC

So we get PQ || AC and

PQ = ½ AC …… (1)

Consider △ BCD

We know that Q and R are the midpoints of BC and CD

So we get QR || BD and

QR = ½ BD ……. (2)

Consider △ ACD

We know that S and R are the midpoints of AD and CD

So we get RS || AC and

RS = ½ AC …….. (3)

Consider △ ABD

We know that P and S are the midpoints of AB and AD

So we get SP || BD and

SP = ½ BD ……. (4)

Consider all the equations

PQ || RS and QR || SP

Hence, PQRS is a parallelogram

It is given that AC = BD

It can be written as

½ AC = ½ BD

So we get

PQ = QR = RS = SP

We know that AC and BD intersect at point O

So we get PS || BD

PN || MO

Based on equation (1)

We get PQ || AC

PM || NO

We know that the opposite angles are equal in a parallelogram

∠ MPN = ∠ MON

We know that ∠ BOA = ∠ MON

So we get

∠ MPN = ∠ BOA

We know that AC ⊥ BD and ∠ BOA = 90^o

So we get

∠ MPN = 90^o

It can be written as

∠ QPS = 90^o

We know that PQ = QR = RS = SP

Therefore, it is proved that PQRS is a square.