Question

PQRS is a parallelogram. M and N are mid-points of PQ and QR. Diagonals PR and QS meet at O.

1. Prove that MONQ is a parallelogram.
2. If PQ = 8 cm and PS = 6 cm, find the perimeter of MONQ.

Answer 1

Given:

• PQRS is a parallelogram,
• M and N are midpoints of sides PQ and QR, respectively,
• Diagonals PR and QS meet at point O.

We are tasked with proving that MONQ is a parallelogram and then finding the perimeter of MONQ if PQ = 8 cm and PS = 6 cm.

Step 1: Prove that MONQ is a parallelogram

To prove that MONQ is a parallelogram, we need to show that opposite sides of quadrilateral MONQ are parallel and equal in length.

Use the properties of midpoints and diagonals:

• In parallelogram PQRS, the diagonals PR and QS bisect each other at point O, meaning O is the midpoint of both diagonals. 
• Since M and N are the midpoints of sides PQ and QR, respectively, the segment MN joins the midpoints of two sides of triangle PQR. 

Now, consider the following:

• M and N are midpoints and the line MN is parallel to diagonal PR because both lines are parallel to the sides of the parallelogram. 
• The midpoint of PR is O and since the diagonals bisect each other, we conclude that line ON is parallel to line QS.

Since opposite sides MN || ON and MQ || NQ are parallel and equal in length, quadrilateral MONQ satisfies the conditions of a parallelogram. 

Thus, we have proven that MONQ is a parallelogram.

Step 2: Find the perimeter of MONQ 

Length of sides of MONQ: Since M and N are midpoints of sides PQ and QR of parallelogram PQRS, the segment MN is parallel to PR and is half its length (by the midpoint theorem). Similarly, segment MQ is parallel to diagonal QS and is also half its length.

Given:

• PQ = 8 cm
• PS = 6 cm

In parallelogram PQRS, the diagonals bisect each other, so:

• PR = PQ = 8 cm
• QS = PS = 6 cm

The lengths of the sides of parallelogram MONQ are half the lengths of the diagonals of PQRS, so:

`MN = 1/2 xx PR = 1/2 xx 8 = 4  cm`

`MQ = 1/2 xx QS = 1/2 xx 6 = 3  cm`

Since opposite sides of a parallelogram are equal, the perimeter of MONQ is

Perimeter of MONQ = 2 × (MN + MQ)

= 2 × (4 + 3)

= 2 × 7

= 14 cm