Home
Class 12
MATHS
If D, E and F be the middle points of th...

If D, E and F be the middle points of the sides BC,CA and AB of the `DeltaABC`, then `AD+BE+CF` is

A

A. a zero vector

B

B. a unit vector

C

C. 0

D

D. none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`AD=OD-OA`
`=(b+c)/(2)-a=(b+c-2a)/(2)` [where, O is the origin for reference]
Similarly, `BE=OE-OB=(c+a)/(2)-b`
`=(c+a-2b)/(2)`
and `CF=(a+b-2b)/(2)`

Now, `AD+BE+CF=(b+c-2a)/(2)+(c+a-2b)/(2)+(a+b-2c)/(2)=0`
Promotional Banner

Similar Questions

Explore conceptually related problems

If D , E and F are the mid-points of the sides BC , CA and AB respectively of the DeltaABC and O be any point, then prove that OA+OB+OC=OD+OE+OF

If P and Q are the middle points of the sides BC and CD of the parallelogram ABCD, then AP+AQ is equal to

If D,E and F are the mid-points of the sides BC,CA and AB, respectively of a DeltaABC and O is any point, show that (i) AD+BE+CF=0 (ii) OE+OF+DO=OA+OB+OC

If D, E and F are three points on the sides BC, CA and AB, respectively, of a triangle ABC such that the lines AD, BE and CF are concurrent, then show that " "(BD)/(CD)*(CE)/(AE)*(AF)/(BF)=-1

If D, E and F are respectively, the mid-points of AB, AC and BC in DeltaABC , then BE + AF is equal to

D,E and F are respectively the mid-points of the sides BC, CA and AB of triangle ABC show that (i) BDEF is a parallelogram. (ii) ar (DEF) = 1/4 ar (ABC) (iii) ar (BDEF) = 1/2 ar (ABC)

In DeltaABC, D, E and F are the midpoints of sides AB, BC and CA respectively. Show that DeltaABC is divided into four congruent triangles, when the three midpoints are joined to each other. (ΔDEF is called medial triangle)

D, E and F are respectively the mid-points of sides AB, BC and CA of Delta ABC . Find the ratio of the areas of Delta DEF and Delta ABC .

D,Eand F are respectively the mid - points of sides AB, BC and CA of DeltaABC . Find the ratio of the areas of DeltaDEF and DeltaABC.

D, E and F are respectively the mid-points of the sides AB, BC and CA of a ΔABC. Prove that by joining these mid-points D, E and F, the ΔABC is divided into four congruent triangles