One interior angle of a rhombus is 60° and the length of its shorter diagonal is 8 cm. What is the area of the rhombus in square centimetres?

Introduction / Context:
This problem tests your understanding of the geometry of a rhombus when one interior angle and one diagonal are known. A rhombus has four equal sides, and its diagonals intersect at right angles. When one angle is 60 degrees, there are special relationships between the side length and the diagonals which allow us to compute the area efficiently without first finding all interior details separately.

Given Data / Assumptions:

• The quadrilateral is a rhombus.
• One interior angle of the rhombus is 60 degrees.
• The shorter diagonal has length 8 cm.
• All sides of the rhombus are equal in length.
• Area of a rhombus can be found using the formula Area = (d1 * d2) / 2, where d1 and d2 are the diagonals.

Concept / Approach:
In a rhombus, if the interior angle between two sides is 60 degrees, the diagonals are related to the side length a by special formulas. For such a rhombus, one diagonal equals a and the other diagonal equals a * square root of 3. In this question the shorter diagonal is given as 8 cm, so that diagonal must equal a. Once we know the side a, we immediately know the other diagonal and can calculate the area using the diagonal product formula.

Step-by-Step Solution:
Let the side length of the rhombus be a cm. For a rhombus with interior angle 60 degrees, the diagonals are a and a * square root of 3. The shorter diagonal is given as 8 cm, so a = 8 cm. Longer diagonal d2 = a * square root of 3 = 8 * square root of 3. Area of rhombus = (d1 * d2) / 2. Substitute d1 = 8 and d2 = 8 * square root of 3. Area = (8 * 8 * square root of 3) / 2 = (64 * square root of 3) / 2 = 32 * square root of 3 sq cm.

Verification / Alternative check:
Another way is to split the rhombus into four congruent right triangles by its diagonals. Each triangle has sides related to the side of the rhombus, and using trigonometry you again get one diagonal equal to a and the other equal to a * square root of 3. This confirms that using a = 8 cm gives the correct area 32 * square root of 3 square centimetres.

Why Other Options Are Wrong:
64√3 sq cm: This treats the product of the diagonals as the area without dividing by 2. 32√2 sq cm and 64√2 sq cm: These use square root of 2 instead of square root of 3 and do not match the geometry of a 60 degree rhombus. 48√3 sq cm: This comes from an incorrect assumption about diagonal lengths or an arithmetic slip.

Common Pitfalls:
Forgetting that the area formula for a rhombus with diagonals uses division by 2. Using generic formulas that relate diagonals to sides without accounting for the specific 60 degree angle. Assuming the given diagonal is the longer one rather than the shorter one.

Final Answer:
The area of the rhombus is 32√3 sq cm