An equilateral triangle has height 9 cm. What is the radius, in centimetres, of the circumcircle that passes through its three vertices?

Introduction / Context:
This problem involves an equilateral triangle and its circumcircle. The height of the triangle is given, and we are asked to find the radius of the circle that passes through all three vertices, known as the circumradius. The question tests familiarity with standard formulas for equilateral triangles and the relationship between side length, height, and circumradius.

Given Data / Assumptions:

• The triangle is equilateral with side length a.
• The height (altitude) of the triangle is given as 9 cm.
• All internal angles are 60 degrees.
• We use the standard formula for the height of an equilateral triangle: h = (sqrt(3) / 2) * a.
• We also use the formula for the circumradius of an equilateral triangle: R = a / sqrt(3).

Concept / Approach:
The main idea is to first determine the side length a of the equilateral triangle using the given height. Once a is known, the circumradius R can be found using the formula R = a / sqrt(3). Both formulas are derived from simple right triangle relationships in an equilateral triangle, where the height splits the triangle into two congruent right triangles.

Step-by-Step Solution:
Step 1: Use the height formula for an equilateral triangle: h = (sqrt(3) / 2) * a.Step 2: Substitute h = 9 cm: 9 = (sqrt(3) / 2) * a.Step 3: Solve for a: a = 9 * 2 / sqrt(3) = 18 / sqrt(3).Step 4: Rationalise if desired: a = (18 * sqrt(3)) / 3 = 6 * sqrt(3).Step 5: Use the formula for circumradius of an equilateral triangle: R = a / sqrt(3).Step 6: Substitute a = 6 * sqrt(3): R = (6 * sqrt(3)) / sqrt(3) = 6 cm.

Verification / Alternative check:
We can check consistency by recalling another relationship: in an equilateral triangle, the height h is related to the circumradius by h = (3 / 2) * R.Substitute R = 6 cm into this relationship: h = (3 / 2) * 6 = 9 cm, exactly matching the given height.This confirms that the computed circumradius is correct.

Why Other Options Are Wrong:
A value of 3 cm is too small and would give a height of only 4.5 cm if used in the relation h = (3 / 2) * R.A value of 9 cm makes the height equal to 13.5 cm from the same relation, which does not match the given data.Similarly, 12 cm gives an even larger height and is inconsistent with the provided triangle height.Only 6 cm satisfies all the geometric relationships.

Common Pitfalls:
A common mistake is to confuse the inradius with the circumradius.Another error is to misremember the height formula, sometimes writing h = a * sqrt(3) instead of h = (sqrt(3) / 2) * a.Errors in algebraic manipulation when solving for a can also lead to incorrect values for the circumradius.

Final Answer:
The radius of the circumcircle is 6 cm.