In geometry (equilateral triangles and medians), the length of a median of an equilateral triangle is given as 12√3 cm. Using this information, determine the exact area of the equilateral triangle in square centimetres.

Introduction / Context:
This problem tests properties of an equilateral triangle, specifically the relationship between its side length, height, and median, and how that relationship leads to the area. In an equilateral triangle, the median, altitude, and angle bisector from any vertex coincide as a single segment. Competitive exams often include such questions to check whether a learner remembers standard formulas and can apply them correctly to compute area from partial information.

Given Data / Assumptions:
- The triangle is equilateral, so all three sides are equal in length.
- The length of a median (which is also the height) is 12√3 cm.
- We need to find the area of the triangle in square centimetres.
- Standard formulas for equilateral triangles and right triangles apply.

Concept / Approach:
In an equilateral triangle of side a, the altitude or median from any vertex to the opposite side has length (√3/2) * a. We can invert this formula to find the side length when the median is known. Once the side a is found, the area of an equilateral triangle is given by (√3/4) * a^2. The question is essentially a two step substitution problem using these standard geometric results.

Step-by-Step Solution:
1) Let the side of the equilateral triangle be a cm. Then the median (also the height) is h = (√3/2) * a. 2) We are told that h = 12√3, so (√3/2) * a = 12√3. 3) Divide both sides by √3 to get a/2 = 12, which gives a = 24 cm. 4) Use the area formula for an equilateral triangle: Area = (√3/4) * a^2. 5) Substitute a = 24: Area = (√3/4) * 24^2 = (√3/4) * 576. 6) Simplify: 576/4 = 144, so Area = 144√3 square centimetres.

Verification / Alternative check:
We can confirm the side length by forming a right triangle made from half of the equilateral triangle. Half the base is a/2 = 12, and the height is 12√3. By the Pythagoras relation, a^2 = (a/2)^2 + h^2 = 12^2 + (12√3)^2 = 144 + 432 = 576, giving a = 24, which matches our earlier result. Substituting back again produces 144√3, confirming the calculation.

Why Other Options Are Wrong:
Option A (288 sq. cm) and Option B (144 sq. cm) ignore the √3 factor required in the area formula of an equilateral triangle. Option C (288√3 sq. cm) doubles the true area, which might arise from mis using the altitude formula. Option E (96√3 sq. cm) results from an incorrect intermediate side length. Only Option D correctly matches the exact area based on the given median length.

Common Pitfalls:
Many students confuse the role of the median with a general median in scalene triangles and forget that in an equilateral triangle it is also the altitude. Others mistakenly use 1/2 * base * height but insert the median value without first verifying the correct base, especially when they do not remember the altitude side relationship. Another common error is missing or misplacing the √3 factor. Keeping a clear list of standard formulas for equilateral triangles helps avoid these mistakes.

Final Answer:
Using the relation between the median and side length in an equilateral triangle and then applying the area formula, the area of the triangle is 144√3 sq. cm.