In a rhombus ABCD, each side has length 12 cm and one interior angle is 120°. What is the length of its longer diagonal (in centimetres)?

Introduction / Context:
This problem concerns a rhombus, a special type of parallelogram with all sides equal. It asks for the length of the longer diagonal when the side length and one interior angle are known. The question tests knowledge of how diagonals in a rhombus relate to side lengths and interior angles, and it requires either use of trigonometry or a derived formula for diagonals based on the included angle between sides.

Given Data / Assumptions:

• ABCD is a rhombus.
• Each side has length 12 cm.
• One interior angle, say angle A, is 120°.
• We need the length of the longer diagonal.

Concept / Approach:
In a rhombus, the diagonals bisect each other at right angles in general only for a square, but more importantly here, each diagonal bisects the angles at its endpoints. If we denote side length by a and the included angle between two sides by θ, formulas for the diagonals can be derived using the law of cosines. The squared lengths are:

• d₁^2 = a^2 + a^2 − 2 * a * a * cosθ = 2a^2 (1 − cosθ)
• d₂^2 = a^2 + a^2 − 2 * a * a * cos(180° − θ) = 2a^2 (1 + cosθ)

One diagonal corresponds to the angle θ and the other to its supplement. For θ = 120°, cos120° = −1/2, which simplifies these expressions.

Step-by-Step Solution:
Step 1: Let side length a = 12 cm and let the given interior angle be θ = 120°. Step 2: Compute cos120°. We know cos120° = cos(180° − 60°) = −cos60° = −1/2. Step 3: Use the formula for one diagonal: d₁^2 = 2a^2 (1 − cosθ). Step 4: Substitute a = 12 and cosθ = −1/2: d₁^2 = 2 * 12^2 * (1 − (−1/2)) = 2 * 144 * (1 + 1/2). Step 5: Evaluate (1 + 1/2) = 3/2, so d₁^2 = 2 * 144 * 3/2 = 144 * 3 = 432. Step 6: Therefore d₁ = √432 = √(144 * 3) = 12√3 cm. Step 7: For completeness, compute the other diagonal using d₂^2 = 2a^2 (1 + cosθ) = 2 * 144 * (1 − 1/2) = 288 * 1/2 = 144, so d₂ = 12 cm. Step 8: Comparing 12 cm and 12√3 cm, the longer diagonal is clearly 12√3 cm.

Verification / Alternative check:
Diagonal d₂ is opposite the obtuse angle 120°, and diagonal d₁ is opposite the acute angle 60°. Since the diagonal opposite the acute angles tends to be longer, it is consistent that d₁ = 12√3 is the longer diagonal. The numerical value 12√3 is approximately 20.78 cm, which is greater than 12 cm, confirming that we selected the correct diagonal as “longer.”

Why Other Options Are Wrong:
6√3 cm and 6√2 cm are clearly too small, being less than the side length itself. 12√2 cm is between 12 cm and 12√3 cm, and it does not satisfy either of the diagonal length formulas derived from the law of cosines for θ = 120°. Only 12√3 cm is consistent with the geometry of a rhombus with side 12 cm and interior angle 120°.

Common Pitfalls:
A common mistake is to assume that diagonals of any rhombus are perpendicular and apply right triangle relations directly, which is only always true for a square and for a rhombus with special angles. Another pitfall is to forget that one diagonal corresponds to angle θ and the other to 180° − θ when applying the law of cosines. Some students also misremember cos120° as +1/2 instead of −1/2. Carefully using the correct trigonometric values and formulas ensures the right answer.

Final Answer:
The length of the longer diagonal of the rhombus is 12√3 cm.