Parallelogram from sides and a diagonal — find area via angle: A parallelogram has adjacent sides 60 m and 40 m, and one diagonal is 80 m. Find its area (in sq m).

Introduction / Context:
Area of a parallelogram is a*b*sinθ. With a diagonal known, the interior angle θ can be recovered using the law of cosines on the diagonal triangle.

Given Data / Assumptions:

• a = 60 m, b = 40 m
• One diagonal p = 80 m
• Law of cosines for diagonal: p^2 = a^2 + b^2 + 2ab*cosθ (for one diagonal)

Concept / Approach:
Compute cosθ from the diagonal formula, then sinθ = sqrt(1 − cos^2θ), and finally area = a*b*sinθ.

Step-by-Step Solution:
a^2 + b^2 = 3600 + 1600 = 5200cosθ = (p^2 − a^2 − b^2)/(2ab) = (6400 − 5200)/(2*60*40) = 1200/4800 = 1/4sinθ = sqrt(1 − (1/4)^2) = sqrt(15/16) = sqrt(15)/4Area = 60*40*(sqrt(15)/4) = 600*sqrt(15) sq. m

Verification / Alternative check:
The other diagonal would correspond to the minus sign in the cosine law; area stays identical for the same θ.

Why Other Options Are Wrong:
480 and 320 assume sinθ = 0.2 or smaller; 450√15 is not supported by the cosine-derived angle.

Common Pitfalls:
Mistaking which diagonal formula sign to use or skipping the sine computation.

Final Answer:
600√ 15 sq. m