Given :
ABCD is a quadrilateral, where P, Q, R and S are the mid points of the sides AB, BC, CD and DA.
Mid-point theorem : The line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.
(i) In △ ADC,
S and R are the mid-points of side AD and CD respectively.
By mid-point theorem,
⇒ SR || AC .......(1)
⇒ SR = ......(2)
Hence, proved that SR || AC and SR = .
(ii) In Δ ABC, P and Q are mid-points of sides AB and BC.
By using the mid-point theorem,
⇒ PQ || AC .........(3)
⇒ PQ = ......(4)
From equations (3) and (4), we get :
⇒ PQ = SR
Hence, proved that PQ = SR.
(iii) From equation (1) and (3), we get :
⇒ PQ || AC || SR
⇒ PQ || SR
Also,
⇒ PQ = SR (Proved above)
We know that,
If one pair of opposite sides are equal and parallel, then the figure is parallelogram.
Hence, proved that PQRS is a parallelogram.
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)