NCERT Solutions
Class 9 Maths
Areas of Parallelograms and Triangles
Ex.9.3 Q.5
D, E and F are respectively the mid-points of the sides BC, CA and AB of a ∆ ABC. Show that:
1. BDEF is a parallelogram.
2. ar (DEF) = 
3. ar (BDEF) =
Given: D, E and F are the mid-points of the sides BC, CA and AB of ΔABC.
1. In ΔABC,
F is the mid-point of side AB and E is the mid-point of side AC.
So, EF || BC
In a triangle, the line segment joining the mid-points of any two sides is parallel to the third
side.
So, EF || BD ………... (1)
Similarly, ED || BF ……. (2)
From equation (1) and (2), we get
BDEF is a parallelogram.
[a quadrilateral is a parallelogram if opposite sides are parallel]
2. As in (1), we can prove that
AFDE and FDCE are parallelograms.
FD is a diagonal of the parallelogram BDEF.
So, area(ΔFBD) = area(ΔDEF) …………… (3)
Similarly, area(ΔDEF) = area(ΔEAE) …………… (4)
and area(ΔDEF) = area(ΔDCE) …………… (5)
From equation (3), (4) and (5), we get
area(ΔFBD) = area(ΔDEF) = area(ΔFAE) = area(ΔDCE) ………… (6)
So, ΔABC is divided into four non-overlapping triangles ΔFBD, ΔDEF, ΔFAE and ΔDCE.
Hence, area(ΔABC) = area(ΔFBD) + area(ΔDEF) + area(ΔFAE) + area(ΔDCE) + area(ΔDEF)
=> area(ΔABC) = 4 × area(ΔDEF)
[From equation 6]
=> area(ΔDEF) = 
……... (7)
(iii) area (BDEF) = area(ΔFBD) + area(ΔDEF)
[From equation 3]
= area(ΔDEF) + area(ΔDEF)
= 2 × area(ΔDEF)
= 2 ×
= 