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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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