Question

P is any point on the side BC of a ∆ABC. P is joined to A. If D and E are the midpoints of the sides AB and AC respectively and M and N are the midpoints of BP and CP respectively then quadrilateral DENM is
(a) a trapezium
(b) a parallelogram
(c) a rectangle
(d) a rhombus

Answer 1

Given: In ∆ABC, M, N, D and E are the mid-points of BP, CP, AB and AC, respectivley.

In ∆ABP,

$\because$ D and M are the mid-points of AB,and BP, respectively. (Given)

$\therefore$ BM = $\frac{1}{2}$AP and BM || AP (Mid-point theorem) ...(i)

Again, in ∆ACP,

$\because$ E and N are the mid-points of AC,and CP, respectively. (Given)

$\therefore$ EN = $\frac{1}{2}$AP and EN || AP (Mid-point theorem) ...(ii)

From (i) and (ii), we get

BM = EN and BM || EN

But this a pair of opposite sides of the quadrilateral DENM.

So, DENM is a parallelgram.

Hence, the correct option is (b).