prove that if a line divides any two side of a triangle in the same ratio, the line must be parallel to the third side.
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Step-by-step explanation:
Given : The line l intersects the sides PQ and side PR of ΔPQR in the points M and N respectively such that
MQ
PM
=
NR
PN
and P−M−Q, P−N−R.
To Prove : Line l ∥ Side QR
Proof : Let us consider that line l is not parallel to the side QR. Then there must be another line passing through M which is parallel to the side QR.
Let line MK be that line.
Line MK intersects the side PR at K, (P−K−R)
In ΔPQR, line MK∥ side QR
∴
MQ
PM
=
KR
PK
....(1) (B.P.T.)
But
MQ
PM
=
NR
PN
....(2) (Given)
∴
KR
PK
=
NR
PN
[From (1) and (2)]
∴ KR
PK+KR
=
NR
PN+NR
(P−K−R and P−N−R)
∴ the points K and N are not different.
∴ line MK and line MN coincide
∴ line MN∥ Side QR
Hence, the converse of B.P.T. is proved.
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