Given :
ABCD is a parallelogram, where E and F are the mid-points of sides AB and CD.
We know that,
Opposite sides of a parallelogram are equal and parallel.
∴ AB || CD and AB = CD
∴ AE || FC
Since, AB = CD
Dividing both sides of equation by 2, we get :
⇒ =
⇒ AE = FC
∴ AEFC is a parallelogram.
∴ AF || CE
From figure,
⇒ PF || QC
⇒ EQ || AP
In △ DQC,
F is the mid-point of DC and FP || CQ.
By converse of mid-point theorem, we get :
P is the mid-point of DQ.
⇒ DP = PQ .....(1)
In △ APB,
E is the mid-point of AB and EQ || AP.
By converse of mid-point theorem, we get :
Q is the mid-point of BP.
⇒ PQ = QB ......(2)
From equations (1) and (2) we get :
⇒ PQ = QB = DP.
∴ AF and EC trisect BD.
Hence, proved that AF and EC trisect the diagonal BD.
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)