Home
Class 12
MATHS
If D,E and F are the mid-points of the s...

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

Text Solution

Verified by Experts


(i) `AD+BE+CF==(d-a)+(e-b)+(f-c)`
`=(d+e+f)-(a+b+c)=0` [usingg eq. (i)]
`implies AD+BE+CF=0`
(ii) Here, `OE+OF+OD=e+f-d`
`=(c+a)/(2)+(a+b)/(2)-(b+c)/(2)=a=OA`
`therefore OE+OF+OD=OA`
Promotional Banner

Topper's Solved these Questions

  • VECTOR ALGEBRA

    ARIHANT MATHS|Exercise Exercise For Session 1|7 Videos

  • VECTOR ALGEBRA

    ARIHANT MATHS|Exercise Exercise For Session 2|17 Videos

  • TRIGONOMETRIC FUNCTIONS AND IDENTITIES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|18 Videos

Similar Questions

Explore conceptually related problems

In the fig D,E and F are the mid points of the sides BC,CA and AB respectively of triangleABC . If BE and DF intersect at X while CF and DE ntersect at Y. prove that XY=1/4BC

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

D,E and F are mid-points of the sides BC, CA and AB respectively of triangleABC . Prove that ar(||gmBDEF)= 1/2 ar (triangleABC)

D and E are the mid-points of the sides AB and AC respectively of DeltaABC . DE is produced to F. To prove that CF is equal and parallel to DA, we need an additional information which is

E and F are the mid points of non-parallel sides AD and BC respectively of a trapezium prove that EF||AB.

ABCD is a quadrilateral in which P,Q,R and S are the mid points of sides AB,BC,CD and DA respectively AC is a diagonal show that: PQ=SR

D,E and F are mid-points of the sides BC, CA and AB respectively of triangleABC . Prove that ar(triangleDEF) = 1/4 ar (triangleABC)

Let the area of a given triangle ABC be Delta . Points A_1, B_1, and C_1 , are the mid points of the sides BC,CA and AB respectively. Point A_2 is the mid point of CA_1 . Lines C_1A_1 and A A_2 meet the median BB_1 points E and D respectively. If Delta_1 be the area of the quadrilateral A_1A_2DE , using vectors or otherwise find the value of Delta_1/Delta

Dand E are the mid-points of the sides AB and AC of DeltaABC and O is any point on side BC. O is joined to A. If P and Q are the mid-points of OB and OC respectively, then DEQP is

D and E are points on sides AB and AC respectively of a DeltaABC such that ar (DeltaBCE) = ar (DeltaBCD) . Show that DE || BC .