Question

P is any point on the side BC of a $\Delta ABC$. 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 ,
∵ D and M are the mid-points of AB,and BP, respectively.
∴ BM = 1/2AP and BM || AP
(Mid-point theorem)
(Given)-----(i)
Again, in ∆ ACP,
∵ E and N are the mid-points of AC ,and CP, respectively.
∴ EN = 1/2AP and EN || AP
(Mid-point theorem)
(Given)-----(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).