P,Q,R are respectively the mid points of sides BC,CA and AB of atriangle ABC.PR and BQ meet at X..CR and PQ meet at Y.Prove that XY=1/4 BC.
Given:P,Q,R are the mid points of sides BC,CA and AB of a triangle ABC. PR and BQ meet at X. CR and PQ meet at Y.
Proof: In Δ ABC, Q and R are the mid-points of sides AC and AB respectively.
⇒QR = BP [P is the mid-point of BC].
Using Mid-point Theorem, it can also be said that QR || BC.
⇒ QR || BP
In quadrilateral BPQR, BP || QR and BP = QR.
Thus, BPQR is a parallelogram
Now, the diagonals BQ and PR of the parallelogram BPQR bisect each other at X.
Thus, X is the mid-point of PR
Similarly, it can be formed that Y is the mid-point of PQ
In ΔPQR, X and Y are the mid-points of sides PR and PQ respectively
Here is the proof to your question.