# Prove that the Quadrilateral Formed by Joining the Mid-points of Consecutive Sides of a Rhombus is a Rectangle. - Mathematics

Sum

Prove that the quadrilateral formed by joining the mid-points of consecutive sides of a rhombus is a rectangle.

#### Solution

In ΔABC, P and Q are mid points of sides AB and BC respectively.

Therefore, PQ || AC and PQ = (1)/(2)"AC"(using mid-point theorem) ...(1)

R and S are the mid points of CD and AD respectively

Therefore, RS || AC and RS = (1)/(2)"AC"(using mid-point theorem) ...(2)

From equations (1) and (2), we have
PQ || RS and PQ = RS
As in quadrilateral PQRS one pair of opposite sides are equal and parallel to each other, so, it is a parallelogram.
Let diagonals of rhombus ABCD intersects each other at point O.
MQ || ON ...(PQ || AC)
QN || OM ...(QR || BD)
So, OMQN is parallelogram
∠MQN = ∠NOM
∠PQR = ∠NOM
But, ∠NOM = 90° ...(diagonals of a rhombus are perpendicular to each other)
∠PQR = 90°
Clearly PQRS is a parallelogram having one of its interior angle as 90°.
Hence, PQRS is rectangle.

Is there an error in this question or solution?
Chapter 19: Quadrilaterals - Exercise 19.2

#### APPEARS IN

Frank Class 9 Maths ICSE