Question

ABCD is a kite having AB = AD and BC = CD. Prove that the figure formed by joining the mid-points of the sides, in order, is a rectangle.

Answer 1

ABCD is a kite such that and

Quadrilateral PQRS is formed by joining the mid-points P,Q,R and S of sides AB,BC,CD and AD respectively.

We need to prove that Quadrilateral PQRS is a rectangle.

In , P and Q are the mid-points of AB and BC respectively.

Therefore,

and

Similarly, we have

and

Thus,

and

Therefore, PQRS is a parallelogram.

Now,

But, P and S are the mid-points of AB and AD

…… (I)

In : P and S are the mid-point of side AB and AD

By mid-point Theorem, we get:

Or,

In , P is the mid-point of side AB and

By Using the converse of mid-point theorem, we get:

M is the mid-point of AO

Thus,

…… (II)

In and , we have:

(Common)

[From (I)]

[From (II)]

By SSS Congruence theorem, we get:

By corresponding parts of congruent triangles property, we get:

But,

and

Therefore,

(, Corresponding angles should be equal)

Or,

We have proved that

Similarly, .

Then we can say that and

Therefore, is a parallelogram with

Or, we can say that is a rectangle.

We get:

Also, PQRS is a parallelogram.

Therefore, PQRS is a rectangle.

Hence proved.