The smaller diagonal of a rhombus is equal to the length of each of its sides. If each side is 6 cm long, what is the area of an equilateral triangle whose side is equal to the larger diagonal of the rhombus?

Introduction / Context:
This problem links properties of a rhombus with those of an equilateral triangle. Rhombus diagonals and equilateral triangle area formulas are standard mensuration topics. Here, you must first use the rhombus geometry to find the larger diagonal, then treat that diagonal as the side of an equilateral triangle and compute its area.

Given Data / Assumptions:
• Each side of the rhombus is 6 cm. • The smaller diagonal of the rhombus is equal to the side length, so smaller diagonal d1 = 6 cm. • The larger diagonal d2 is unknown. • An equilateral triangle has its side equal to d2, and we are asked to find its area.

Concept / Approach:
In a rhombus, all sides are equal and the diagonals intersect at right angles and bisect each other. If the diagonals are d1 and d2, then each half diagonal has lengths d1 / 2 and d2 / 2. A side of the rhombus forms the hypotenuse of a right triangle whose legs are these half diagonals. Thus, we can use Pythagoras theorem to find d2. Once we know d2, we use the formula for the area of an equilateral triangle with side a: area = (sqrt(3) / 4) * a^2.

Step-by-Step Solution:
1. Let side of the rhombus be a = 6 cm. 2. Smaller diagonal is d1 = 6 cm, so half of it is d1 / 2 = 3 cm. 3. Let d2 be the larger diagonal; half of it is d2 / 2. 4. In one of the right triangles formed by the diagonals, the legs are d1 / 2 and d2 / 2, and the hypotenuse is the side a. 5. Apply Pythagoras theorem: a^2 = (d1 / 2)^2 + (d2 / 2)^2. 6. Substitute a = 6 and d1 / 2 = 3: 6^2 = 3^2 + (d2 / 2)^2. 7. Compute: 36 = 9 + (d2^2 / 4). 8. Rearranging: d2^2 / 4 = 36 - 9 = 27. 9. Multiply by 4: d2^2 = 108. 10. Hence d2 = sqrt(108) = sqrt(36 * 3) = 6 * sqrt(3). 11. Now, an equilateral triangle has side a_equilateral = d2 = 6 * sqrt(3) cm. 12. Area of an equilateral triangle with side a is (sqrt(3) / 4) * a^2. 13. So area = (sqrt(3) / 4) * (6 * sqrt(3))^2 = (sqrt(3) / 4) * (36 * 3) = (sqrt(3) / 4) * 108. 14. Simplify: 108 / 4 = 27, so area = 27 * sqrt(3) square centimetres.

Verification / Alternative check:
You can check the reasonableness by noting that d2 is longer than the side of the rhombus, which is consistent with a longer diagonal. The equilateral triangle with side 6 * sqrt(3) is significantly larger than a triangle with side 6, and its area should be larger than 6^2. Since 27 * sqrt(3) is approximately 27 * 1.732, which is around 46.8, it fits a realistic area for such a side length.

Why Other Options Are Wrong:
• 18√3 sq.cm: This corresponds to a smaller effective side length and does not match the calculated d2. • 32√3 sq.cm: This does not follow from the computed d2^2 = 108 and would require a different side. • 36√3 sq.cm: This would correspond to an equilateral triangle with even larger side than 6 * sqrt(3). • 24√3 sq.cm: Again, this value does not come from the formula with a^2 = 108.

Common Pitfalls:
Common mistakes include assuming the diagonals of a rhombus are equal (which is only true for a square), or forgetting that the diagonals are perpendicular and bisect each other. Some students incorrectly use the entire diagonals instead of half diagonals when applying Pythagoras theorem. Being careful with halves and the right triangle structure is essential.

Final Answer:
The area of the equilateral triangle is 27√3 sq.cm.