Question

ABCD is a rectangle and P,Q, R and S are mid POINT of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus ?

Given

• ABCD is a rectangle
• P, Q, R and S are midpoint of the sides AB, BC , CD and DA

Required to Find

To show that the quadrilateral PQRS is a rhombus

Solution

Here we join A and C in triangle ABC.

P is the mid point of AB

Q is the mid ppint of BC

PQ || AC

( Line segments JOINING the mid points of two sides of a triangle ia parallel to AC and also is half of it )

In triangle ADC

R is mid point of CD

S is mid point of AD

RS || AC

( line segments joining the mid points of two sides of a triangle is parallel to THIRD side and also is half of it )

So,

PQ || RS and PQ = RS

( one pair of opposite side is parallel and equal )

In triangle APS and triangle BPQ

AP = BP ( P is the mid point of AB )

angle PAS = angle PBQ

( all the angles of rectangle are 90° )

AS = BQ

triangle APS congruent to triangle BPQ ( SAS test )

PS = PQ

BS = PQ and PQ = RS

( opposite sides of paralleogram is equal )

PQ = RS = PS = RQ ( all sides are equal )

therefore, PQRS is a paralleogram with all sides equal. .

So, PQRS is a rhombus