Question

In TRIANGLE ABC;D,E,F are mid points of side AB,BC and CA respectively.If ar (ABC)=64 cm2, then ar (DEF) in cm2:
A)16
B)4
C)256
D)32​

Answer 1

Answer:

c)256

Step-by-step explanation:

Answer 2

Answer:

A) 16 cm²

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 :-

$\frac{area of triangle DFE}{area of triangle ABC} =\frac {DE^2}{AC^2}$

Since, DECF is a parallelogram. So DE = FC

On substituting:

$\frac{area of triangle DFE}{area of triangle ABC} =\frac{FC^2}{AC^2}$

Also, F is the midpoint of AC

So AC=2*FC

$\frac{Area of triangle DFE}{Area of triangle of ABC} =\frac{FC^2}{(2FC)^2}$

$\frac{Area of triangle DFE}{Area of triangle of ABC} =\frac{1}{4}$

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

So, Ar(triangle DEF)/Ar(triangle ABC)=1/4

Then, as Ar(triangle ABC)=64cm² so,

Ar(triangle DEF)=64/4=16cm²