*
In A ABC, D, E, F are midpoints of sides AB, BC and CA respectively. If
arva ABC) = 64 cm", then ar(A DEF) in cm2 :
[ ]
(4 16
(B) 4
(D)
32
Answers
Answer:
Step-by-step explanation:
In the given question, we know Triangle DEF formed with midpoints is similar to the Outer Triangle ABC
On the basis of the similarity, we can say,
If two triangles are similar then the ratio of their area is equal to the square of the ratio of their corresponding sides
Mathematically can be written as :-
\frac{area of triangle DFE}{area of triangle ABC} =\frac {DE^2}{AC^2}
areaoftriangleABC
areaoftriangleDFE
=
AC
2
DE
2
Since, DECF is a parallelogram. So DE = FC
On substituting:
\frac{area of triangle DFE}{area of triangle ABC} =\frac{FC^2}{AC^2}
areaoftriangleABC
areaoftriangleDFE
=
AC
2
FC
2
Also, F is the midpoint of AC
So AC=2*FC
\frac{Area of triangle DFE}{Area of triangle of ABC} =\frac{FC^2}{(2FC)^2}
AreaoftriangleofABC
AreaoftriangleDFE
=
(2FC)
2
FC
2
\frac{Area of triangle DFE}{Area of triangle of ABC} =\frac{1}{4}
AreaoftriangleofABC
AreaoftriangleDFE
=
4
1
Hence ratio of area of triangle DEF and triangle ABC is given as:
Ratio of area of triangle DEF : area of triangle ABC = 1 : 4 answer a 16