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if a line divides any two sides of a triangle in the same ratio, then prove that line is parallel to the third side. |
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Answer» Answer: 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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