P, Q and R are, respectively, the mid-points of sides BC, CA and AB of a triangle ABC. PR and BQ meets at X. CR and PQ meets at Y. Prove that:
XY = 1/4 BC
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Answers
hey....your answer:-
What is your question?
P,Q and R are respectively the mid points of sides BC,CA and AB of triangle ABC.PR and BQ meet atX. 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 = \frac{1}{4} 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 = \frac{1}{2} 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.
\implies 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 = \frac{1}{2} RQ
From 3,
XY = \frac{1}{2} + \frac{1}{2} BC
XY = \frac{1}{4} BC
Hence Proved.
⛄ GIVEN:-
A ∆ABC with P, Q and R as the mid-points of BC, CA and AB respectively. PR and BQ meet at X and CR and PQ meet at Y.
⛄ CONSTRUCTION:-
Join X and Y.
⛄ PROOF:-
Since the line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it, therefore, Q and R are the mid-points of AC and AB respectively.
Therefore,
RQ || BC and RQ = 1/2 BC ......→ (i)
=> RQ || BP and RQ=BP [Since P is the mid-point of BC. Therefore, 1/2 BC = BP]
=> BPQR is a parallelogram.
Now,
Since the diagonals of a parallelogram bisect each other,
Therefore,
X is the mid-point of PR [Since X is the point of intersection of diagonals BQ and PR of ||gm BPQR]
Similarly, Y is the mid-point of PQ.
Now,
Consider ∆PQR. Here, XY is the line segment joining the mid-points of sides PR and PQ.
Therefore,
XY = 1/2 RQ
But,
RQ = 1/2 BC [From (i) ]
=> XY = 1/4 BC.
⛄ HENCE PROVED!!!
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