The diagonals of a rhombus measure 8 cm and 6 cm respectively. What is the area of this rhombus in square centimetres?

Introduction / Context:
This question focuses on the special area formula for a rhombus in terms of its diagonals. A rhombus is a quadrilateral with all sides equal, and its diagonals have the useful property of being perpendicular bisectors of each other. In many exam problems, diagonals are provided instead of base and height, and you are expected to apply the appropriate formula quickly and accurately.

Given Data / Assumptions:

• The figure is a rhombus with four equal sides.
• The lengths of its diagonals are 8 cm and 6 cm.
• The diagonals intersect at right angles and bisect each other.
• We must compute the area in square centimetres.
• Standard Euclidean geometry formulas for quadrilaterals apply.

Concept / Approach:
The area A of a rhombus can be expressed in multiple ways. In terms of base and height, it is A = base * height. In terms of its diagonals d1 and d2, the more convenient formula is A = (1 / 2) * d1 * d2. This formula results from the fact that the diagonals divide the rhombus into four congruent right triangles, each with legs equal to half of each diagonal. Using this relation, we can directly substitute the diagonal lengths and compute the area in one simple step.

Step-by-Step Solution:
Let the diagonals be d1 = 8 cm and d2 = 6 cm.Use the area formula for a rhombus: A = (1 / 2) * d1 * d2.Substitute the values: A = (1 / 2) * 8 * 6.Compute the product: 8 * 6 = 48.So A = (1 / 2) * 48 = 24 sq cm.

Verification / Alternative check:
To verify, imagine the diagonals intersecting at right angles and dividing the rhombus into four right triangles. Each triangle has legs 4 cm and 3 cm (half of each diagonal). The area of one such triangle is (1 / 2) * 4 * 3 = 6 sq cm. Since there are four congruent triangles, total area = 4 * 6 = 24 sq cm. This alternative reasoning perfectly matches the formula based method, confirming the result is correct.

Why Other Options Are Wrong:
Option 96 sq cm is obtained by multiplying the diagonals directly without taking half, which overestimates the area by a factor of 2. Option 60 sq cm and 48 sq cm may come from random combinations of 8 and 6 with approximate heights or mistaken formulas. Option 36 sq cm corresponds to using 6 * 6 instead of 8 * 6, which ignores one diagonal entirely. None of these match the correct formula A = (1 / 2) * d1 * d2.

Common Pitfalls:
A frequent mistake is to forget the factor of 1 / 2 and simply multiply the diagonals. Some test takers confuse the rhombus formula with that of a rectangle or square, for which area is side squared or length times breadth. Others may incorrectly think that diagonals must be equal, which is true for squares but not for all rhombuses. Recognizing the shape and recalling the specific formula avoids such confusion.

Final Answer:
The area of the rhombus is 24 sq cm.