Question

In the adjoining figure, G is the centroid

of DEF. H is a point on ray DG such that

D - G - H and DG = GH.

Prove that GEHF is a parallelogram.
​

Answer 1

Given:-

Point G (centroid) is the point of concurrence of the medians of ADEF.

DG = GH

To prove:-

□GEHF is a parallelogram.

Proof:-

Let ray DH intersect seg EF at point I such that E-I-F.

∴ seg DI is the median of ∆DEF.

∴ El = FI ……(i)

Point G is the centroid of ∆DEF.

∴ DG/GI = 2/1 [Centroid divides each median in the ratio 2:1]

∴ DG = 2(GI)

∴ GH = 2(GI) [DG = GH]

∴ GI + HI = 2(GI) [G-I-H]

∴ HI = 2(GI) – GI ∴ HI = GI ….(ii)

From (i) and (ii), GEHF is a parallelogram [A quadrilateral is a parallelogram, if its diagonals bisect each other]