The area of a rhombus is 54 square centimetres and the length of one diagonal is 9 centimetres. Find the length of its other diagonal in centimetres.

Introduction / Context:
This question is about a rhombus, a special type of quadrilateral where all sides are equal. The diagonals of a rhombus have special properties and are often used to find its area or missing dimensions. Here you must use the relationship between diagonals and area.

Given Data / Assumptions:

• The figure is a rhombus.
• Area of the rhombus = 54 square centimetres.
• One diagonal, say d1, is 9 cm.
• We must find the other diagonal, d2, in centimetres.

Concept / Approach:
For a rhombus with diagonals d1 and d2:
Area = (1 / 2) * d1 * d2.
This formula arises because the diagonals of a rhombus are perpendicular and bisect each other, effectively splitting it into four right angled triangles. We substitute the known area and one diagonal to solve for the second diagonal.

Step-by-Step Solution:
Step 1: Use the formula Area = (1 / 2) * d1 * d2. Step 2: Let d1 = 9 cm and d2 be the unknown diagonal. Step 3: Substitute values: 54 = (1 / 2) * 9 * d2. Step 4: Simplify (1 / 2) * 9 = 4.5, so 54 = 4.5 * d2. Step 5: Solve for d2: d2 = 54 / 4.5. Step 6: Compute 54 / 4.5 = 12. Step 7: Therefore, the other diagonal is 12 cm long.

Verification / Alternative check:
We can substitute back to confirm: Area = (1 / 2) * 9 * 12 = (1 / 2) * 108 = 54 square centimetres, which matches the given area. This confirms the correctness of d2 = 12 cm.

Why Other Options Are Wrong:
6 cm: Would give area (1 / 2) * 9 * 6 = 27 square centimetres, which is too small.
9 cm: Would give area (1 / 2) * 9 * 9 = 40.5 square centimetres, still not 54.
18 cm: Produces area (1 / 2) * 9 * 18 = 81 square centimetres, which is larger than given.
24 cm: Produces area (1 / 2) * 9 * 24 = 108 square centimetres, double the required area.

Common Pitfalls:
Students sometimes confuse the rhombus area formula with that of a rectangle or parallelogram (base * height). They may also forget the factor of 1 / 2 or misidentify the given length as a side rather than a diagonal. Always confirm that you are dealing with diagonals and use the proper formula.

Final Answer:
The length of the other diagonal is 12 cm.