We know that,
The diagonals of a rectangle are equal.
⇒ BD = AC = x (let)
In △ ABC,
P and Q are the mid-points of AB and BC respectively.
By mid-point theorem,
⇒ PQ || AC and PQ = ....(1)
In △ ADC,
S and R are the mid-points of AD and CD respectively.
SR || AC and SR = .....(2)
From equation (1) and (2), we get :
PQ || SR and PQ = SR
In quadrilateral PQRS, one pair of opposite sides are equal and parallel to each other.
∴ PQRS is a parallelogram.
In △ BCD, Q and R are the mid-points of side BC and CD respectively.
By mid-point theorem,
⇒ QR || BD and QR = ....(3)
In △ BAD, P and S are the mid-points of side AB and AD respectively.
By mid-point theorem,
PS || BD and PS = .......(4)
From equations (1), (2), (3), (4), we get :
PQ = QR = SR = PS
Hence, proved that the quadrilateral PQRS is a rhombus.
Quadrilaterals — Interactive Study Guide
Parallelogram Properties
Quick test: To check if a quadrilateral is a parallelogram, verify ANY ONE of these (or show one pair of opposite sides is both equal AND parallel).
Mid-Point Theorem
The line joining mid-points of two sides of a triangle is parallel to the third side and half its length.
Quick Self-Check
- Angle sum of a quadrilateral? (360°)
- ABCD is a parallelogram, ∠A = 75°. Find ∠B, ∠C, ∠D. (105°, 75°, 105°)
- In ΔPQR, M and N are midpoints of PQ and PR. QR = 12 cm. Find MN. (6 cm)