P,Q,R are respectively the mid points of sides BC,CA and AB of atriangle ABC PR and BQ meet at X - Maths - Quadrilaterals

Question

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

Gursheen kaur. · Accepted Answer

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 To Prove: $X Y = \frac{1}{4} B C$ Proof: In Î” ABC, Q and R are the mid-points of sides AC and AB respectively $∴ Q R = \frac{1}{2} B C (i)$

â‡’QR = BP [P is the mid-point of BC] Using Mid-point Theorem, it can also be said that QR || BC â‡’ 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 Î”PQR, X and Y are the mid-points of sides PR and PQ respectively $∴ X Y = \frac{1}{2} Q R \Rightarrow X Y = \frac{1}{2} \times \frac{1}{2} \times B C [U \sin g (i)] \Rightarrow X Y = \frac{1}{4} \times B C$ Hence Proved

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.