18. P, Q & Rare respectively, the mid points of sides BC, CA & AB of a

1.	18. P, Q & Rare respectively, the mid points of sides BC, CA & AB of a triangle ABC. PR & BQmeet at XăCR & PQ meet at Y. Prove that XY--BCéŁ
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 = 1/4BC

18. P, Q & Rare respectively, the mid points of sides BC, CA & AB of a triangle ABC. PR & BQmeet at XăCR & PQ meet at Y. Prove that XY--BCéŁ

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 = 1/4BC