Question

If D,E,F are respectively the mid-point of the side BC,CA,AB of a triangle ABC,then the ratio of the area of triangle DEF to the area of triangle ABC is?
​

Answer 1

Step-by-step explanation:

ANSWER

Given: In ΔABC, D,E and F are midpoints of sides AB,BC and CA respectively.

BC=EC

Recall that the line joining the midpoints of two sides of a triangle is parallel to third side and half of it.

Therefore,we have:

DF=d

2

1

BC

⇒

BC

DF

=

2

1

....(1)

AC

DE

=

2

1

....(2) and

AB

EF

=

2

1

....(3)

From (1), (2) and (3) we have

But if in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar

Therefore, ΔABC∼ΔEDF [By SSS similarity theorem]

Hence area of ΔABC: area of ΔDEF=4:1