M and N divide the side of AB of triangle ABC into 3 equal parts Line segments MP and - Maths - Quadrilaterals

Question

M and N divide the side of AB of triangle ABC into 3 equal parts
Line segments MP and NQ are both parallel to BC and meets at AC at
P and Q respectively Prove P and Q divide AC into 3 equal parts

Harleen Birdi · Answer

Hi !!!
Here is the answer to your question
<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/editlive_temp/7%20png">
Given : A triangle ABC in which M and N divides
AB into three equal parts i e AM = MN = NB and MP || NQ || BC
To prove: P and Q divide AC into three equal
parts i e AP = PQ = QC
Construction Join MC Let it intersects QN at
R
Proof: We have AM = MN = NB
â‡’ AM = MN
â‡’ M is the mid point of AN
Consider Î” ANQ,
M is the mid point of AN and MP || NQ (Given)
âˆ´ By mid point theorem, P is the mid point of
AQ
â‡’ AP = PQ (1)
Consider Î”BCM, N is the mid point of MB (as MN =
NB) and NR || BC (as NQ || BC)
â‡’ by mid point theorem , R is the mid point
of MC
Now, Consider Î” CPM,
R is mid point of MC ( proved above) and QR || PM (as QN ||
PM)
â‡’ Q is mid point of CP (by mid point
theorem)
â‡’ CQ = QP
From (1) and (2)
AP = PQ = QC
â‡’ P and Q divides AC into 3 equal parts
Hence, the result is proved
Best wishes !!!

M and N divide the side of AB of triangle ABC into 3 equal parts. Line segments MP and NQ are both parallel to BC and meets at AC at P and Q respectively. Prove P and Q divide AC into 3 equal parts.