Question

In parallelogram PQRS, PQ = 25 cm, QR = 39 cm and diagonal PR = 56 cm. Find its area.

Answer 1

Given:

• Parallelogram PQRS
• PQ = 25 cm
• QR = 39 cm
• Diagonal PR = 56 cm

Step wise calculation:

1. Using the cosine rule to find the angle θ between sides PQ and QR:

PR² = PQ² + QR² – 2 × PQ × QR × cos(θ)

56² = 25² + 39² – 2 × 25 × 39 × cos(θ)

3136 = 625 + 1521 – 1950 cos(θ)

1950 cos(θ) = 2146 – 3136 = –990

`cos(θ) = (-990)/1950 = -0.50769`

2. Then,

`sin(θ) = sqrt(1 - cos^2(θ))`

= `sqrt(1 - (-0.50769)^2)`

= `sqrt(1 - 0.2577)`

= `sqrt(0.7423) ≈ 0.86144`

3. Area is:

Area = PQ × QR × sin(θ)

= 25 × 39 × 0.86144

= 839.3 cm²

4. Rounding to the nearest whole number yields:

840 cm²