An equilateral triangle ABC is inscribed in a circle of radius 20√3 cm. What is the length (in cm) of each side of the equilateral triangle?

Introduction / Context:
This question checks your understanding of the relation between the side length of an equilateral triangle and the radius of the circumcircle in which it is inscribed. It is a standard geometry result that appears frequently in aptitude and competitive exams.

Given Data / Assumptions:

• Triangle ABC is equilateral.
• Triangle ABC is inscribed in a circle, so all three vertices lie on the circle.
• The radius R of the circle (circumradius) is 20√3 cm.
• We need to find the side length a of the equilateral triangle.
• We use standard Euclidean geometry and exact values (no rounding until the end if needed).

Concept / Approach:
For an equilateral triangle of side a, the relationship between the side length and circumradius R is a well known formula: R = a / √3 This can be derived from splitting the equilateral triangle into two 30 60 90 right triangles by drawing a median. Once we know this relation, we can solve for a in terms of R: a = R * √3 Substituting R = 20√3 cm will give the required side length.

Step-by-Step Solution:
Step 1: Let a be the side length of the equilateral triangle ABC. Step 2: Use the formula for circumradius of an equilateral triangle: R = a / √3. Step 3: Rearrange the formula to express a: a = R * √3. Step 4: Substitute R = 20√3 cm. Step 5: So a = 20√3 * √3 = 20 * 3 = 60 cm. Step 6: Therefore, each side of triangle ABC is 60 cm long.

Verification / Alternative check:
We can confirm by checking the reverse relation. If a = 60 cm, then the circumradius R should be: R = a / √3 = 60 / √3 = (60√3) / 3 = 20√3 cm. This matches the given radius, so the answer 60 cm is fully consistent.

Why Other Options Are Wrong:
30 cm: If a = 30 cm, then R = 30 / √3 = 10√3 cm, which is half of the given radius.
40 cm: For a = 40 cm, R = 40 / √3, which is not equal to 20√3 cm.
50 cm: For a = 50 cm, R = 50 / √3, which again does not match 20√3 cm.
20√3 cm: This option confuses side length with radius. 20√3 cm is the radius, not the side length.

Common Pitfalls:
Many students mistakenly use formulas for inradius instead of circumradius, or they forget the special relation of equilateral triangles and try to apply general triangle formulas. Another common error is mishandling square roots, such as writing (√3) * (√3) as √6 instead of 3. Keeping the formula R = a / √3 clearly in mind helps to avoid these mistakes.

Final Answer:
The side length of the equilateral triangle is 60 cm.