Question

18. 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 perpendicular to BD then prove
that the quadrilateral formed is a square.​

Answer 1

As shown in the figure, let P, Q, R, S be the midpoints of the sides A, BC, CD and AD respectively.

Also, AC = BD, AC ⊥ BD.

To prove:

Quadrilateral PQRS form is a square.

Proof:

In Δ ABC,

PQ║AC

∴      [Mid point theorem]     [1]

In Δ ACD,

SR║AC

∴      [Mid point theorem]      [2]

So, PQRS is a parallelogram.

In BCD,

QR║BD

∴      [Mid point theorem]      [3]

From Eq (2) and (3), we get

SR║AC, QR║BD  

AC ⊥ BD       [given]

∴ SR ⊥ QR                           [4]

So, PQRS is a parallelogram is a square.

Also, AC = BD       [given]

Dividing both the sides by 2, we get

SR = QR          [from Eq (2) and (3)]     [5]

Hence from Eq (4) and (5),

Quadrilateral PQRS form is a square.

Answer 2

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 = 90o So we get ∠ MPN = 90⁰

It can be written as ∠ QPS = 90⁰

We know that PQ = QR = RS = SP

Therefore, it is proved that PQRS is a square.