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Hi Shivkumar,

The text version of the answer 

PQRS  is a parallelogram in which A is the mid point of SR and B is a point on PR , such that BR = 1/4 PR if AB is produced  it meets QR , at C  prove that C is a mid point of QR

Given: PQRS is a parallelogram

    A is mid-point of SR

    and BR = 1/4 PR

To prove: C is mid-point of QR

Construction: Join SQ

Proof:

Since Diagonals of a parallelogram bisect each other,

OP = OR = 1/2 PR

But, BR = 1/4 PR

∴ B is mid-point of OR

Now, in Δ SOR,

A is mid-point of SR,

B is mid-point of OR

By converse of Basic Proportionality Theorem,

If line a divides any two side of a triangle in the same ratio, then the line is parallel to third side. 

∴   AB ∥ OS

Thus, we can say that

  AC ∥ SQ

Now,

In Δ SRQ

A is mid-point of SR and AC ∥ SQ

By Basic Proportionality Theorem,

If a line is drawn parallel to one side of a triangle to intersect the other two side in distinct points, the other two sides are divided in the same ratio.

∴   QC = RC

⇒ C is mid-point of QR

Hence proved