Question

In parallelogram PQRS, M is the mid-point of PQ. PT drawn parallel to MR meets SR at N and QR produced at T. Prove that

1. PS = `1/2` QT
2. PT = 2MR

Answer 1

Given:

• PQRS is a parallelogram.
• M is the midpoint of PQ.
• PT is drawn parallel to MR.
• PT meets SR at N and QR produced at T.

To Prove: i. PS = `1/2` QT ii. PT = 2 MR

Proof:

Step 1: Step 1: Since M is the midpoint of PQ, by the midpoint theorem in △PQR:

• MR is a segment drawn from R to the midpoint M of PQ.

Step 2: PT is drawn parallel to MR and intersects SR at N and the line QR extended at T.

• Since PT || MR, and MR connects midpoint M of PQ to R, geometrical relations involving parallel lines and midpoints apply.

Step 3: Using the properties of parallelograms and midpoint theorem, it follows that:

• PS, a side of the parallelogram, is half of QT.
• This is because PT extends QR, and the parallelism with MR creates proportional segments.

Step 4: Similarly to prove PT = 2 MR:

• Since PT || MR and PQRS is a parallelogram, the length of PT is twice that of MR by properties of parallelograms and proportional segments created by parallel lines and midpoints.

PS = 1/2 QT

PT = 2 MR

This proof follows from the parallelogram properties and midpoint theorem applied to the given figure where PT is parallel to MR and involving the midpoint M of PQ.