Question

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

Answer 1

$\textbf{ Hello Mate! }$

Given : P, Q and R are mid-points in triangle ABC.

To prove : XY = 1 BC / 4

Proof : Since P,Q and R are mid points so

QR || BC => QR || BP

PQ || AB => PQ || BR

Hence, BPQR is parallelogram.

As diagonals of parallelogram bisect each other, so BX = XQ. Hence, X is mid-point on BQ.

Similarily, PCQR is parallelogram.

Hence, Y is mid point on RC.

As, X and Y are midpoints so XY || QR and XY = 1 QR / 2

Moreover, QR || BC ( Q and R are mid-points ). Hence, QR = 1 BC / 2

So, XY = 1/2 ( 1 BC / 2 )

XY = 1 BC / 4

Hence proved!

Have great future ahead!

Answer 2

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

Step-by-step explanation:

I hope it helps you

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