Question

In the given figure T is the exterior point on the diagonal PR of a parallelogram PQRS . SR produced meets OT at N and QR produced meets ST at M. Prove that MN||SQ ​

Answer 1

Given :-

• T is the exterior point on the diagonal PR of a parallelogram.

• SR produced meets OT at N and QR produces meets ST at M

Solution :-

Here,

In ΔPQT, RN || PQ

[ T is the exterior point on the diagonal PR of a parallelogram and SR produced OT at N ]

Therefore,

TN / NQ = TR / RP. ( 1 )

[ If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio ]

Now,

In ΔSPT, MR||SP

[ T is the exterior point on the diagonal PR of a parallelogram and QR produced meets ST at M]

Therefore,

TR / RP = TM / MS. ( 2 )

[ If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio]

From ( 1 ) and ( 2 )

TN / NQ = TM/MS

In ΔTSQ

TN / NQ = TM / MS

Therefore,

MN || SQ

[ If a line divides any two sides of triangle in the same ratio, then the line parallel to the third side ]

Hence, Proved

Theorem kept in mind :-

• If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points. The other two sides are divided in the same ratio.

• If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side $.$

Answer 2

Given :-

• T is the exterior point on the diagonal PR of a parallelogram.

• SR produced meets OT at N and QR produces meets ST at M

Solution :-

Here,

In ΔPQT, RN || PQ

[ T is the exterior point on the diagonal PR of a parallelogram and SR produced OT at N ]

Therefore,

TN / NQ = TR / RP. ( 1 )

[ If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio ]

Now,

In ΔSPT, MR||SP

[ T is the exterior point on the diagonal PR of a parallelogram and QR produced meets ST at M]

Therefore,

TR / RP = TM / MS. ( 2 )

[ If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio]

From ( 1 ) and ( 2 )

TN / NQ = TM/MS

In ΔTSQ

TN / NQ = TM / MS

Therefore,

MN || SQ

[ If a line divides any two sides of triangle in the same ratio, then the line parallel to the third side ]

Hence, Proved

Theorem kept in mind :-

• If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points. The other two sides are divided in the same ratio.

• If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side ..