Question

points p, q and r are the mid point of side BC,CA and AB Respectively of ∆ABC and they form ∆PQR. X is the point intersection of line segments PR and BQ. Y is the point of intersection line segments CR and PQ.prove that. XY=1/4BC.​

Answer 1

P,Q and R are respectively the mid points of sides BC, CA and AB of triangle ABC. PR and BQ meet at X. CR and PQ meet at Y. prove that XY=1/4BC

Answer

Given

ABC is a Triangle.

P is the m.p of BC

Q is the m.p of CA

R is the m.p of AB

To prove

XY = BC

Proof

In ΔABC

R is the midpoint of AB.

Q is the midpoint of AC.

∴ By Midpoint Theorem,

RQ║BC

RQ║BP → 1 [Parts of Parallel lines]

RQ = BC → 2

Since P is the midpoint of BC,

RQ = BP → 3

From 1 and 3,

BPQR is a Parallelogram.

BQ and PR intersect at X

Similarly,

PCQR is a Parallelogram.

PQ and CR intersect at Y.

X and Y are Midpoints of sides PR and PQ respectively.

In ΔPQR

X is the midpoint of PR

Y is the midpoint of PQ

∴ By Midpoint Theorem,

XY = RQ

From 3,

XY = + BC

XY = BC