What is the area (in square centimetres) of a rhombus whose diagonals measure 12 cm and 14 cm respectively?

Introduction / Context:
A rhombus is a special quadrilateral in which all four sides are equal, but the angles are not necessarily right angles. One of the most useful formulas for a rhombus relates its area to the lengths of its diagonals. Since many aptitude questions provide diagonal lengths instead of base and height, being comfortable with this formula helps in solving such problems quickly and reliably.

Given Data / Assumptions:

• We have a rhombus.
• The lengths of its diagonals are 12 cm and 14 cm.
• We are asked to find the area of this rhombus in square centimetres.
• No additional special conditions are given, so we use the standard diagonal area formula.

Concept / Approach:
In a rhombus, the diagonals intersect at right angles and bisect each other. The area of a rhombus can be computed using the formula:
Area = (d1 * d2) / 2 where d1 and d2 are the lengths of the diagonals. This formula comes from the fact that the four right triangles formed by the diagonals can be grouped into two congruent rectangles or into one full parallelogram whose area equals half the product of the diagonal lengths.

Step-by-Step Solution:
Step 1: Identify the diagonal lengths as d1 = 12 cm and d2 = 14 cm. Step 2: Recall the formula Area = (d1 * d2) / 2. Step 3: Substitute the given values into the formula: Area = (12 * 14) / 2. Step 4: First compute the product of the diagonals: 12 * 14 = 168. Step 5: Divide by 2: Area = 168 / 2 = 84. Step 6: Therefore, the area of the rhombus is 84 square centimetres.

Verification / Alternative check:
We can think of each diagonal as being split in half at the intersection point, giving segments of 6 cm and 7 cm. These segments form right triangles with legs 6 and 7. The area of one such triangle is (1/2) * 6 * 7 = 21 sq cm. There are four congruent triangles inside the rhombus, so the total area is 4 * 21 = 84 sq cm. This confirms the earlier computation using the formula.

Why Other Options Are Wrong:

• 42: This is exactly half of the correct area, which would result from mistakenly using Area = d1 * d2 / 4.
• 168: This is the product of the diagonal lengths without dividing by 2 and therefore overestimates the area.
• 63 and 56: These values do not follow from any correct application of the rhombus diagonal relationships and are included as distractors.

Common Pitfalls:
A common mistake is to confuse the formula for the area of a rectangle (length * breadth) with that of a rhombus and to skip the division by 2. Another error is to misread or miscalculate the diagonals. Always check that you multiply the diagonals correctly and then divide the result by 2 to obtain the actual area.

Final Answer:
Hence, the area of the rhombus is 84 sq cm.