A rhombus has all its sides equal to 10 cm, and one of its diagonals is 12 cm long. Using these measurements, what is the area (in cm²) of the rhombus?

Introduction / Context:
This problem concerns a rhombus, a special type of quadrilateral in which all four sides are equal and the diagonals intersect at right angles and bisect each other. We are given the length of each side and one diagonal and must find the area of the rhombus. The key relationships here are between the side length, the diagonals, and the right triangles formed when the diagonals intersect. Using these relationships, we can find the second diagonal and then apply the standard area formula for a rhombus.

Given Data / Assumptions:

• Each side of the rhombus = 10 cm.
• One diagonal (say d1) = 12 cm.
• The diagonals of a rhombus are perpendicular and bisect each other.
• We must find the area of the rhombus in square centimetres.

Concept / Approach:
In a rhombus, the diagonals intersect at right angles and bisect each other. So each half of a diagonal forms a right triangle with half of the other diagonal and the side of the rhombus. If d1 and d2 are the diagonals, then (d1 / 2)^2 + (d2 / 2)^2 = side^2. We can use this relation to find the unknown diagonal d2. Once both diagonals are known, the area of the rhombus is given by (1 / 2) * d1 * d2.

Step-by-Step Solution:
Let the side of the rhombus be a = 10 cm.Let the diagonals be d1 and d2, with d1 = 12 cm and d2 unknown.Half of d1 is d1 / 2 = 12 / 2 = 6 cm.Half of d2 is d2 / 2.In one of the right triangles formed by the diagonals, the hypotenuse is the side a and the legs are d1 / 2 and d2 / 2.So (d1 / 2)^2 + (d2 / 2)^2 = a^2.Substitute values: 6^2 + (d2 / 2)^2 = 10^2.That is 36 + (d2^2 / 4) = 100.Rearrange: d2^2 / 4 = 100 − 36 = 64.So d2^2 = 64 * 4 = 256, giving d2 = 16 cm.Area of the rhombus = (1 / 2) * d1 * d2.So area = (1 / 2) * 12 * 16 = 6 * 16 = 96 cm².

Verification / Alternative check:
We can verify by checking the right triangle formed by half diagonals. With d1 = 12 and d2 = 16, half lengths are 6 and 8. Then the side of the rhombus should be √(6^2 + 8^2) = √(36 + 64) = √100 = 10 cm, which matches the given side. This confirms that d2 = 16 cm is correct and ensures that the area calculation based on these diagonals is accurate.

Why Other Options Are Wrong:
The value 48 cm² would result from mistakenly multiplying 12 and 8 without using the half factor correctly. The values 144 cm² and 192 cm² are too large and typically arise from misapplying formulas or ignoring the halving when computing diagonals. The value 120 cm² does not follow from any consistent application of the rhombus area formulas. Only 96 cm² is consistent with both the diagonal relationship and the correct area formula (1 / 2) * d1 * d2.

Common Pitfalls:
Many students forget that the diagonals are bisected, and therefore use full diagonal lengths in the Pythagoras theorem rather than half lengths, leading to incorrect values. Another mistake is to mix up the formulas for the area of a rhombus and a general quadrilateral. Some also bypass the diagonal relation and attempt to use base-height formulas without a clear known height, which complicates the problem unnecessarily. Sticking to the diagonal properties makes this problem straightforward.

Final Answer:
The area of the rhombus is 96 cm².