Given:
- P, Q, R, S are respectively the midpoints of the sides of parallelogram ABCD.
- The area of parallelogram PQRS is 32.5 sq.cm.

To find:
The area of parallelogram ABCD.

Solution:

1. Understanding the problem:
Let's begin by understanding the given information and the properties of a parallelogram.

- A parallelogram is a four-sided polygon in which opposite sides are parallel and equal in length.
- The midpoints of the sides of a parallelogram divide it into four smaller congruent parallelograms.
- The area of a parallelogram can be calculated using the formula: Area = base × height.

2. Identifying key points and variables:
Let's assign some variables to the given information to make it easier to work with:
- Let the length of side AB be 'a'.
- Let the length of side BC be 'b'.
- Let the length of side CD be 'c'.
- Let the length of side AD be 'd'.

3. Determining the lengths of the sides:
Since P, Q, R, and S are the midpoints of the sides of parallelogram ABCD, we can conclude that:
- PQ = RS = 1/2 AB
- QR = PS = 1/2 BC
- RS = PQ = 1/2 CD
- PS = QR = 1/2 AD

4. Calculating the area of PQRS:
The area of parallelogram PQRS is given as 32.5 sq.cm. Using the formula for the area of a parallelogram, we have:
Area of PQRS = PQ × PS = RS × QR = 32.5 sq.cm

5. Determining the lengths of PQ, PS, RS, and QR:
Since PQ = 1/2 AB and PS = 1/2 AD, we can express PQ and PS in terms of 'a' and 'd':
PQ = 1/2 AB = 1/2 a
PS = 1/2 AD = 1/2 d

Similarly, RS and QR can be expressed in terms of 'b' and 'c' respectively:
RS = 1/2 CD = 1/2 c
QR = 1/2 BC = 1/2 b

6. Solving for the lengths of the sides:
Now, let's substitute the values of PQ, PS, RS, and QR in the area formula:
(1/2 a) × (1/2 d) = (1/2 c) × (1/2 b) = 32.5 sq.cm

Simplifying the equation, we have:
(1/4)ad = (1/4)cb = 32.5 sq.cm

7. Finding the area of ABCD:
Since PQRS is divided into four congruent parallelograms, the area of ABCD can be determined by multiplying the area of PQRS by 4:
Area of ABCD = 4 × 32.5 sq.cm = 130 sq.cm.

8. Conclusion:
The area of parallelogram ABCD is

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Join PR

as ∆PSR and ||gm APRD are on the same base and between same parallels lines
ar(PSR) = 1/2 ar (APRD)

Similarly, ar (PQR) = 1/2ar(PBCR)

ar(PQRS) = ar(PSR) + ar(PQR)
= 1/2ar(APRD) + 1/2ar(PBCR)
= 1/2ar(ABCD)

=> ar(ABCD) = 2 × ar(PORS)
= 2 × 32.5
= 65.0 cm^2

THANK YOU