Question

The mid-points of the sides BC, CA and AB of a ΔABC are D, E and F respectively. Which of the following is true?

• A. Area of (DEF) = $\frac{1}{4}$ Area of $\Delta$ ABC
• B. Area of (BDEF) = $\frac{1}{2}$ Area of $\Delta$ ABC
• C. BDEF is a parallelogram
• D. Area of (BDEF) = $\frac{1}{4}$ Area of $\Delta$ ABC

Answer 1

Answer: (A), (B), (C)

The correct options are
A Area of (DEF) = $\frac{1}{4}$ Area of $\Delta$ ABC
B Area of (BDEF) = $\frac{1}{2}$ Area of $\Delta$ ABC
C BDEF is a parallelogram

E is midpoint of AC (given)
F is midpoint of AB (given)
D is midpoint of BC (given)

EF || BC and EF = $\frac{1}{2}$ BC (mid point theorem)
ED || AB and ED = $\frac{1}{2}$ AB (mid point theorem)
FD || AC and FD = $\frac{1}{2}$ AC (mid point theorem)

ECDF is a ||gm (one pair of opposite sides are equal and parallel)
EDBF is a ||gm (one pair of opposite sides are equal and parallel)
EDFA is a ||gm (one pair of opposite sides are equal and parallel)

area $\Delta$ ECD = area $\Delta$ EDF (diagonal divides a $\Delta$ into two $\Delta$ 's of equal area)
area $\Delta$ DBF = area $\Delta$ EDF
area $\Delta$ AEF = area $\Delta$ EDF
$\therefore$ area $\Delta$ CDE = area $\Delta$ DBF = area $\Delta$ AEF = area $\Delta$ EDF

Hence, area $\Delta DEF=\frac{1}{4}$ area $(\Delta$ ABC)
and area (BDEF) = $\frac{1}{2}$ area $\left(\Delta ABC\right)$