In a ΔABC, D, E, F are the mid-points of sides BC, CA and AB respectively. If ar (ΔABC) = 16cm2, then ar (trapezium FBCE) =
Options
4 cm2
8 cm2
12 cm2
10 cm2
Solution
Given: In ΔABC
(1) D is the midpoint of BC
(2) E is the midpoint of CA
(3) F is the midpoint of AB
(4) Area of ΔABC = 16 cm2
To find: The area of Trapezium FBCE
Calculation: Here we can see that in the given figure,
Area of trapezium FBCE = Area of ||gm FBDE + Area of ΔCDE
Since D and E are the midpoints of BC and AC respectively.
∴ DE || BA ⇒ DE || BF
Similarly, FE || BD. So BDEF is a parallelogram.
Now, DF is a diagonal of ||gm BDEF.
∴ Area of ΔBDF = Area of ΔDEF ……(1)
Similarly,
DE is a diagonal of ||gm DCEF
∴ Area of ΔDCE = Area of ΔDEF ……(2)
FE is the diagonal of ||gm AFDE
∴ Area of ΔAFE = Area of ΔDEF ……(3)
From (1), (2), (3) we have
Area of ΔBDF = Area of ΔDCF = Area of ΔAFE = Area of ΔDEF
But
Area of ΔBDF + Area of ΔDCE + Area of ΔAFE + Area of ΔDEF = Area of ΔABC
∴ 4 Area of ΔBDF = Area of ΔABC
Area of ΔBDF = `1/4` Area of ΔABC
= `1/4 (16)`
= 4 cm2
Area of ΔBDF = Area of ΔDCE = Area of ΔAFE = Area of ΔDEF = 4 cm2 …….(4)
Now
Area of trapezium FBCE = Area of || FBDE + Area of ΔCDE
=(Area of ΔBDF + Area of ΔDEF ) + Area of ΔCDE
= 4 + 4+ 4 (from 4)
= 12 cm2
Hence we get
Area of trapezium FBCE = 12 cm2
