Question

in quadrilateral ABCD , P,Q,R,S are midpoints of AB,BC,CD and DA respectively.Prove that PR and QS bisect eachother.​

Answer 1

Answer:

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Answer 2

Answer:

Given : ABCD is a quadrilateral. P, Q, R and S are the midpoints of AB, BC, CD and DA respectively.

To prove : PR and QS bisect each other.

Construction : Join AC

Solution :

To prove the given condition, we must prove that ABCD is a IIgm.

So,

In triangle ADC, PQ is the midpoint of AD and DQ.  

Thus, PQ || AC                        (by mid-point theorem)

Similarly in triangle ABC,

SR || AC

PQ || AC and SR || AC

Then,

PQ || SR

Similarly, by joining BA, we get,

PS || QR

Then, In quadrilateral PQRS,

PQ || SR and PS || QR

Thus,
PQRS is a ||gm      (Opposite sides are parallel)

Then,

PR and QS bisect each other     (In ||gm, diagonals bisect each other)

HENCE PROVED