Question

D, E and F are the midpoints of the sides BC CA and AB respectively of triangle abc. find ar( triangle DEF / ar( triangle ABC ​

Answer 1

Answer:

Given

D,E and F are respectively the midpoints of sides AB,BC and CA of ΔABC

To find

ar(△ABC)

ar(△DEF)

We know that

The line segment joining the midpoints of any two sides of a triangle is half the third side and parallel to it.

∴FD=

2

1

BC,ED=

2

1

ACandEF=

2

1

AB

In △ABC and △EFD, we have

EF

AB

=

FD

BC

=

ED

AC

=2...(i)

⇒△ABC∼△EFD[by SSS similarity criterion]

Also, We know that

If two triangles are similar, then the ratio of the area of both triangles is equal to the square of the ratio of their corresponding sides

∴

ar(△EFD)

ar(△ABC)

=(

EF

AB

)

2

=4

[from (i)]

⇒

ar(△ABC)

ar(△EFD)

=

4

1

Hence, the ratio of the areas of △DEF and △ABC is 1:4

Answer 2

Step-by-step explanation:

1:4 is the answer I hope it will help you. If you have any doubt in answer you can ask