Ex 8.2, 5
In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.
Given: ABCD is a parallelogram where
E and F are the mid-points of sides
AB and CD respectively
To prove: AF & EC trisect BD
i.e. BQ = QP = DP
Proof: ABCD is a parallelogram.
∴ AB ∥ CD
⇒ AE ∥ CF
& AB = CD
1/2 AB = 1/2 CD
∴ AE = CF
In AECF,
AE ∥ CF & AE = CF
one pair of opposites sides is equal and parallel
∴ AECF is a parallelogram
⇒ AF ∥ CE
∴ PF ∥ CQ & AP ∥ EQ
In ΔDQC,
F is the mid-point of DC
and PF ∥ CQ .
∴ P is the mid-point of DQ.
⇒ PQ = DP
From (1) & (2)
DP = PQ = BQ
Hence, the line segments AF and EC trisect the diagonal BD.
Hence proved

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.