The area of an equilateral triangle is 9√3 sq. cm. What is the height (altitude) of this triangle in centimetres?

Introduction / Context:
This question requires reversing the standard area formula for an equilateral triangle in order to find its side length and then its height. Once the side length is known, the height follows from the basic relation in equilateral triangles. This type of reverse application helps reinforce understanding of formulas rather than only using them in forward direction.

Given Data / Assumptions:

• The triangle is equilateral.
• Area A = 9√3 sq. cm.
• We need the height (altitude) of the triangle.
• All sides and angles are equal, with each angle equal to 60°.
• Standard formulas for equilateral triangles apply.

Concept / Approach:
The area of an equilateral triangle with side length a is A = (√3 / 4) * a^2. Given A, we can solve for a by rearranging this formula. Once a is found, we use the formula for the height h of an equilateral triangle, which is h = (√3 / 2) * a. Substituting the computed side length into this height formula gives the required altitude.

Step-by-Step Solution:
Given A = 9√3 sq. cm and A = (√3 / 4) * a^2.Set (√3 / 4) * a^2 = 9√3.Divide both sides by √3: (1 / 4) * a^2 = 9, so a^2 = 36.Thus side length a = 6 cm.Height h = (√3 / 2) * a = (√3 / 2) * 6 = 3√3 cm.

Verification / Alternative check:
We can verify by recomputing the area from the derived side and height. For a triangle with base a = 6 and height h = 3√3, area should be (1 / 2) * base * height = (1 / 2) * 6 * 3√3 = 9√3 sq. cm. This matches the given area, confirming that both the side and the height are correctly computed.

Why Other Options Are Wrong:
Option 6 cm is the side length, not the height, so it is larger than the correct altitude. Option 6√3 cm is too large and would give a much larger area than 9√3. Option 9 cm would correspond to a different area entirely. Option 4√3 cm would give a base and height combination inconsistent with the given area when substituted into area formulas.

Common Pitfalls:
Many students confuse the side length with the height and may stop after finding a = 6 cm. Others misapply the area formula or forget to multiply by 1 / 4 when isolating a^2. Some misremember the height formula and use a / 2 instead of (√3 / 2) * a, which leads to incorrect results.

Final Answer:
The height of the equilateral triangle is 3√3 cm.