Question

Why basic proportionality theorm is valid only if the line cutting two sides is parallel to the third side? Why not if the line is not parallel?​

Answer 1

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.