In a rhombus, the smaller diagonal is equal to the length of each of its sides. If the length of each side is 4 cm, then the smaller diagonal is 4 cm. Using this information, what is the area (in square centimetres) of an equilateral triangle whose side is equal to the longer diagonal of this rhombus?

Introduction / Context:
This question combines properties of a rhombus with the area formula for an equilateral triangle. It tests understanding of how diagonals relate to the side of a rhombus and how to use these diagonals to derive the side of a new geometric figure, in this case an equilateral triangle whose side equals the longer diagonal of the rhombus.

Given Data / Assumptions:

• The quadrilateral is a rhombus with all four sides equal in length.
• The length of each side is 4 cm.
• The smaller diagonal of the rhombus is equal to the side length, so the smaller diagonal is 4 cm.
• The longer diagonal is unknown and must be calculated.
• An equilateral triangle is constructed with side equal to this longer diagonal, and we must find its area in square centimetres.

Concept / Approach:
For any rhombus, the diagonals are perpendicular bisectors of each other. Each side of the rhombus forms the hypotenuse of a right triangle whose legs are half of each diagonal. If we denote the diagonals by d1 and d2, then side^2 = (d1/2)^2 + (d2/2)^2. After finding the larger diagonal, we use the standard formula for the area of an equilateral triangle: Area = (sqrt(3)/4) * side^2.

Step-by-Step Solution:
Let the smaller diagonal be d1 = 4 cm and the larger diagonal be d2 cm.Half diagonals are d1/2 = 2 cm and d2/2 cm. The side of the rhombus is 4 cm and is the hypotenuse of the right triangle formed by these halves.Use Pythagoras theorem: side^2 = (d1/2)^2 + (d2/2)^2, so 4^2 = 2^2 + (d2/2)^2.This gives 16 = 4 + d2^2/4, so d2^2/4 = 12 and therefore d2^2 = 48. Hence d2 = 4sqrt(3) cm.The equilateral triangle has side a = d2 = 4sqrt(3). Its area is (sqrt(3)/4) * a^2 = (sqrt(3)/4) * 48 = 12sqrt(3) square centimetres.

Verification / Alternative check:
The relationship between side and diagonals in a rhombus is consistent for any rhombus: if one diagonal equals the side, the other diagonal must be longer and determined by Pythagoras theorem. The computed longer diagonal d2 = 4sqrt(3) is greater than 4, which is reasonable. Substituting this value into the equilateral triangle area formula matches the simplified exact value 12sqrt(3). No contradictions arise, confirming the result.

Why Other Options Are Wrong:

• 6 and 12 are too small and ignore the factor of sqrt(3) arising from the equilateral triangle formula.
• 9sqrt(3) would correspond to a smaller effective side length and does not match d2^2 = 48.
• 18 is a plain integer and cannot be the exact area of an equilateral triangle with an irrational side length 4sqrt(3).

Common Pitfalls:

• Forgetting that the diagonals of a rhombus are perpendicular bisectors and misusing Pythagoras theorem.
• Using full diagonals instead of half diagonals when setting up the right triangle relation.
• Applying the wrong area formula for the equilateral triangle or missing the factor sqrt(3)/4.

Final Answer:
12√3