An equilateral triangle has area 9√3 square centimetres. Find the exact height (altitude) of the triangle.

Introduction / Context:Converting between area and altitude in an equilateral triangle relies on two linked formulas: A = (√3/4)a^2 and h = (√3/2)a. Use one to find side length, then compute the altitude.

Given Data / Assumptions:

• Area A = 9√3 cm^2.

Concept / Approach:First find the side a from A = (√3/4)a^2. Then use h = (√3/2)a to get the altitude.

Step-by-Step Solution:

Verification / Alternative check:Compute A from a=6: (√3/4)*36 = 9√3 cm^2; consistent.

Why Other Options Are Wrong:

• 6 cm, 9 cm: These are side-length-like numbers, not the altitude for A = 9√3.
• 6√3 cm: Overshoots by a factor of 2.

Common Pitfalls:Mixing up the area and altitude formulas or skipping the square on a.

Final Answer:3√3 cm