Question

Let P be an interior point of a convex quadrilateral ABCD and K, L, M, N be the midpoints of AB, BC, CD, DA respectively. If Area (PKAN) = 25, Area (PLBK) = 36, and Area (PMDN) = 41, then Area (PLCM) is: -

(a) 20

(b) 29

(c) 52

(d) 54

Answer 1

The correct option is (C) 52.

From mid-point theorem it is given that,

Area(PKAN) = 25 = x + z  ----- (1)

Area(PLBK) = 36 = x + y  ----- (2)

Area(PMDN) = 41 = w + z  ----- (3)

Area(PLCM) = y + w

Now, adding (2) and (3) we get,

⇒  x + y + w + z = 36 + 41

⇒  x + y + w + z = 77 ----- (4)

Subtracting (1) from (4) we get,

⇒ x + y + w + z − x − z = 77 − 25

⇒  y + w = 52

∴  Area of PLCM = 52