P,Q,R are, respectively, the mid points of sides AB, BC and CA and of a triangle ABC. PR and AQ meet at X. BR and PQ meet at Y. Prove that XY = ¼ AB.
Answers
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
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