If D,E,F are respectively the mid points of the sides BC,CA and AB of △ABC and the area of △ABC is 24 sq. cm, then the area of △DEF is -
The correct option is C: 6 cm2
Given: D,E and F are the mid-points of sides BC,AC and AB respectively of ΔABC.
Mid Point Theorem -The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.
FE=12BC [Using above theorem]
FEBC=12
Similarly. DE=12AB [Using above theorem]
DEAB=12
and, FDAC=12
∴FEBC=DEAB=FDAC
Thus, ΔABC∼ΔDEF [by SSS similarity rule]
The ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.
Area ΔDEFArea ΔABC=(DEAB)2
Area ΔDEFArea ΔABC=(DE2 DE)2
Clearly, area (△DEF)=14×area (△ABC)
=(14×24)=6 cm2