A rhombus has diagonals of lengths 12 cm and 14 cm. Using the standard area formula for a rhombus in terms of its diagonals, what is its area (in cm²)?

Introduction: This question checks whether you remember the area formula for a rhombus expressed in terms of its diagonals. A rhombus can be thought of as a special kite where the diagonals are perpendicular bisectors of each other. This gives a convenient formula for its area.

Given Data / Assumptions:

• We have a rhombus.
• Lengths of the diagonals are d₁ = 12 cm and d₂ = 14 cm.
• We must find the area of this rhombus in square centimetres.

Concept / Approach: For any rhombus (and in fact for any kite with perpendicular diagonals), the area can be calculated using the formula: Area = (1/2) * d₁ * d₂. This comes from the fact that the diagonals divide the rhombus into four right triangles of equal area, and their combined area equals half the product of the diagonals.

Step-by-Step Solution: Given d₁ = 12 cm and d₂ = 14 cm. Use the formula: Area = (1/2) * d₁ * d₂. Substitute the values: Area = (1/2) * 12 * 14. Compute 12 * 14 = 168. Now multiply by 1/2: Area = 168 / 2 = 84 cm².

Verification / Alternative check: If you draw the rhombus and its diagonals, you get four congruent right triangles, each with legs 6 cm and 7 cm (half of each diagonal). Area of one triangle is (1/2) * 6 * 7 = 21 cm². Multiply by 4 to get the rhombus area: 4 * 21 = 84 cm², confirming the formula based calculation.

Why Other Options Are Wrong: The value 168 cm² is simply the product of the diagonals, without the necessary factor of 1/2. The values 42 cm² and 63 cm² are partial products that do not correspond to any standard area computation for rhombuses with these diagonals.

Common Pitfalls: A typical mistake is to forget dividing by 2 and use d₁ * d₂ directly. Another is to mix formulas for rectangles or parallelograms with those of rhombuses. Remember that for rhombuses, diagonals and their perpendicular property are a powerful shortcut.

Final Answer: The area of the rhombus is 84 cm².