Question

abcd is a rhombus and p,q,r and s are the mid points of the sides ab,bc,CD and da respectively. show that the quadrilateral pqrs is a rectangle.
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Answer 1

Answer:

Given: ABCD is a rhombus in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA . AC, BD are diagonals.

To prove: the quadrilateral PQRS is a rectangle.

Proof: In

ACD,

S is midpoint of DA. (Given)

R is midpoint of DC. (Given)

By midpoint theorem,

and

...................................1

In

ABC,

P is midpoint of AB. (Given)

Q is mid point of BC. (Given)

By mid point theorem,

and

.................................2

From 1 and 2,we get

and

Thus,

and

So,the quadrilateral PQRS is a parallelogram.

Similarly, in

BCD,

Q is mid point of BC. (Given)

R is mid point of DC. (Given)

By mid point theorem,

So, QN || LM ...........5

LQ || MN ..........6 (Since, PQ || AC)

From 5 and 6, we get

LMPQ is a parallelogram.

Hence,

LMN=

LQN (opposite angles of the parallelogram)

But,

LMN= 90 (Diagonals of a rhombus are perpendicular)

so,

LQN=90

Thus, a parallelogram whose one angle is right angle,ia a rectangle.Hence,PQRS is a rectangle.

Answer 2

Answer:

Rectangle..........