Home
Class 10
MATHS
If D,E and F are the midpoints of sides ...

If D,E and F are the midpoints of sides BC, CA, and Ab respectively of `triangle ABC` then using coordinate geometry prove that Area of `triangle DEF`= `1/4` (Area of `triangle ABC`)

Text Solution

Verified by Experts

let the coordinates of points A, B,C, D,E,F be `(x_1,y_1), (x_2,y_2), (x_3,y_3),((x_1+x_2)/2,(y_1+y_2)/2),((x_1+x_3)/2,(y_1+y_3)/2) , ((x_2+x_3)/2,(y_+y_3)/2)`
`/_ DEF = 1/2*[((x_1+x_2))/2 ((y_1- y_2)/2) + ((x_1+x_3))/2((y_3-y_1)/2) + ((x_1+x_3)/2*(y_2-y_3)/2)]`
`= 1/2*1/4(x_1+x_2)(y_1-y_2) + (x_1+x_3)(y_3-y_1) + (x_2+x_3)(y_2-y_3)|`
by solving we get,
`= 1/2*1/4|x_1(y_3-y_2) + x_2(y_1-y_3) +x_3(y_2-y_1)|`
rearanging the equation by positive sign, we get
`=1/4*1/2|x_1(y_2-y_3) + x_2(y_3-y_1)+ x_3(y_1-y_2)`
= `1/4*` ar`(/_ABC)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If D ,Ea n dF are the mid-points of sides BC, CA and AB respectively of a A B C , then using coordinate geometry prove that Area of △DEF= 1/4 (Area of △ABC)

D,E and F are the mid points of the sides BC, CA and AB respectivley of f triangle ABC then the ratio of the areas of triangle DEF and ABC =………..

If D,E,F are the mid-points of the sides BC,Ca and AB respectively of a triangle ABC ,prove by vector method that Area of Delta DEF=1/4 Area of Delta ABC

D,E,F are the mid points of the sides BC, CA and AB respectively of a triangle ABC . Determine the ratio of the areas of triangle DEF and triangle ABC .

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that: BDEF is a parallelogram.

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that: area of BDEF is half the area of Delta ABC.

If D,E,F are the mid-point of sides AB,BC and CA respectively of triangle then the ratio of the areas of triangles triangleDEF and ABC is

D, E and F are respectively the mid¬ points of the sides BC, CA and AB of triangleABC . Determine the ratio of the areas of triangles DEF and ABC.