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 perpendicular to bd then prove that quadrilateral form is a square

Answer 1

I know it may wrong I also have doubt in same plss anyone help me

Answer 2

Proved below.

Step-by-step explanation:

Given:

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

∴ $PQ= \frac{1}{2} AC$     [Mid point theorem]     [1]

In Δ ACD,

SR║AC

∴ $SR= \frac{1}{2} AC$     [Mid point theorem]      [2]

So, PQRS is a parallelogram.

In BCD,

QR║BD

∴ $QR= \frac{1}{2} BD$     [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

$\frac{1}{2} AC = \frac{1}{2} BD$

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

Hence from Eq (4) and (5),

Quadrilateral PQRS form is a square.