What is the area, in square centimetres, of a rhombus if the lengths of its diagonals are 12 centimetres and 14 centimetres?

Introduction / Context:
This problem focuses on the area formula for a rhombus in terms of its diagonals. Rhombus questions appear frequently because they give practice with a special quadrilateral whose diagonals intersect at right angles and bisect each other. Knowing the diagonal based area formula allows a quick solution without needing side lengths.

Given Data / Assumptions:

• Length of first diagonal d1 = 12 centimetres.
• Length of second diagonal d2 = 14 centimetres.
• We must find the area in square centimetres.
• The figure is a standard rhombus with the usual diagonal properties.

Concept / Approach:
For a rhombus, if d1 and d2 are the lengths of its diagonals, then its area A is given by:
A = (d1 * d2) / 2.
This formula arises because the diagonals of a rhombus are perpendicular and divide it into four right angled triangles. We use the given diagonals directly in this formula to compute the area.

Step-by-Step Solution:
Step 1: Identify the diagonals: d1 = 12 cm and d2 = 14 cm. Step 2: Use the formula A = (d1 * d2) / 2. Compute the product d1 * d2 = 12 * 14. 12 * 14 = 168. Step 3: Now divide by 2: A = 168 / 2 = 84. So the area of the rhombus is 84 square centimetres.

Verification / Alternative check:
We can think of the rhombus as being composed of four right angled triangles, each with legs of length 12/2 = 6 cm and 14/2 = 7 cm. The area of each triangle is (1/2) * 6 * 7 = 21 square centimetres. Four such triangles have total area 4 * 21 = 84 square centimetres. This alternative reasoning matches the result from the diagonal formula, confirming the answer.

Why Other Options Are Wrong:
Option A (42 sq cm): This is exactly half of the correct area and would come from forgetting to multiply the diagonals before dividing by 2. Option B (168 sq cm): This is the product d1 * d2 without dividing by 2, so it overestimates the area. Option D (63 sq cm): This does not match any correct intermediate value and is simply incorrect. Option E (56 sq cm): This could result from a mistaken combination of half of one diagonal with the full other diagonal.

Common Pitfalls:
Students sometimes confuse the formula for the area of a rhombus with that of a parallelogram and use base times height instead. Others may forget the division by 2 in the diagonal based formula. Miscalculating the product 12 * 14 can also lead to errors. Carefully remembering A = (d1 * d2) / 2 and checking multiplication helps avoid these problems.

Final Answer:
The area of the rhombus is 84 sq cm (square centimetres).