Home
Class 10
MATHS
The side B C of a triangle A B C is bise...

The side `B C` of a triangle `A B C` is bisected at `D ;o` is any point in `A D ,B Oa n dC O` produced meet `A Ca n dA B` in `Ea n dF` respectively and `A D` is produced to `X` so that `D` is the mid-point of `O Xdot` Prove that `A O : A X=A F : A B` and show that `F E B Cdot`

Text Solution

Verified by Experts

Join BX and CX.
Since D is the mid -point of BC ( given )
BD= DC …(1)
Also D is the mid-point of OX (given )
OD= DX … (2)
From (1) and (2) we, have
`squareOBXC` is a parallelogram ( diagonals bisect each other)
OC||BX ( opposite sides of a parallelogram)
`Rightarrow FC||BX Rightarrow FO ||BX`
`In DeltaABX, (AO)/(OX)= (AF)/(FB)` ( by B.P theorem) ...(3)
Also , BO||XC ( opposite sides of a parallelogram)
` Rightarrow BE||XCRightarrow OE||XC`
` (AO)/(OX)= (AE)/(EC)` (by B.P theorem) ...(4)
From (3) and (4) , we have
`1 ( AF)/(FB) = (AE) (EC)`
In ` DeltaABC, FE||BC.` (converse of B.P theorem) Hence Proved.
Promotional Banner

Topper's Solved these Questions

  • TRIANGLES

    NAGEEN PRAKASHAN|Exercise Problems From NCERT/ Exemplar|14 Videos

  • TRIANGLES

    NAGEEN PRAKASHAN|Exercise Exercise 6a|23 Videos

  • STATISTICS

    NAGEEN PRAKASHAN|Exercise Revision Exercise Long Answer Questions|5 Videos

  • VOLUME AND SURFACE AREA OF SOLIDS

    NAGEEN PRAKASHAN|Exercise Revisions Exercise Long Answer Questions|5 Videos

Similar Questions

Explore conceptually related problems

The side BC of a triangle ABC is bisected at D;o is any point in AD,BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid-point of OX. Prove that AO:AX=AF:AB and show that FE|BC.

The side BC of a DeltaABC is bisected at D. O is any point on AD, BO and CO are produced to meet AC and AB at E and F respectively and AD is produced to G so that D is the midpoint of OG. Prove that FE|| BC

The side BC of a DeltaABC is bisected at D. O is any point on AD. BO and CO are produced to meet AC and AB at E & F respectively. AD is produced to C1 so that D is the mid-point of OC1. Prove that FE||BC.

In the given figure, side BC of Delta ABC is bisected at D and O is any point AD.BO and CO produced meet AC and AB at E and F respectively, and AD is respectively, and AD is produced to X so that D is the midpoint of OX. Prove that AO:AX=AF:AB and show that EF||BC .

In A A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A Pdot Prove that: a r\ ( R Q C)=3/8\ a r\ (\ A B C)

In A A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A Pdot Prove that: a r\ ( P B Q)=\ a r\ (\ A R C)

In A A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A Pdot Prove that: a r\ (\ P R Q)=1/2a r\ (\ A R C)

A B C D is a parallelogram. A D is a produced to E so that D E=D C and E C produced meets A B produced in Fdot Prove that B F=B Cdot

A B C is a triangle in which D is the mid-point of B C ,\ E\ a n d\ F are mid-points of D C\ a n d\ A E respectively. If area of A B C is 16\ c m^2, find the area of D E F

The diagonals of a rectangle A B C D\ meet at O . If /_B O C=44^0, find /_O A D