Rhombus with one angle 60° and shorter diagonal 8 cm: Find the area of the rhombus (in cm²).

Introduction / Context:A rhombus has all sides equal. Given one interior angle, the diagonals relate to the side and the angle. The area is a*b*sin(θ) where a and b are adjacent sides. With a known diagonal and angle, we can retrieve the side and compute area compactly.

Given Data / Assumptions:

• Interior angle θ = 60°.
• Shorter diagonal d_short = 8 cm.
• All sides equal to a.

Concept / Approach:For a rhombus with side a and angle θ: diagonals are a√(2 + 2cosθ) and a√(2 − 2cosθ). For θ = 60°, cosθ = 1/2. Then the two diagonals are a√3 and a. The shorter diagonal equals a, so a = 8 cm. Area = a^2 * sinθ = 8^2 * sin60° = 64 * (√3/2) = 32√3.

Step-by-Step Solution:

Verification / Alternative check:Rhombus area can also be (d1*d2)/2. Here d1 = a = 8; d2 = a√3 = 8√3 ⇒ area = (8 * 8√3)/2 = 32√3, consistent.

Why Other Options Are Wrong:Values with √2 arise from square/45° assumptions; 64√3 doubles the correct area.

Common Pitfalls:Confusing which diagonal is shorter at θ = 60°; it is a, not a√3.

Final Answer:32√3 sq cm