Difficulty: Easy
Correct Answer: 20
Explanation:
Introduction / Context:
This geometry question involves two important radii associated with an equilateral triangle: the inradius (radius of the inscribed circle) and the circumradius (radius of the circumscribed circle). You are given the inradius and asked to find the circumradius. For equilateral triangles, the relationship between these two radii is particularly simple, which makes the calculation very quick once you know the formula.
Given Data / Assumptions:
Concept / Approach:
For an equilateral triangle with side length a, the inradius r and circumradius R are given by r = (a√3) / 6 and R = (a√3) / 3. These formulas show that the circumradius is exactly double the inradius: R = 2r. Therefore, once the inradius is known, you can immediately find the circumradius without even computing the side length a explicitly.
Step-by-Step Solution:
Verification / Alternative check:
If you want an extra confirmation, you can first compute the side length a from the inradius formula. Since r = (a√3) / 6 and r = 10, we have 10 = (a√3) / 6, so a = 60 / √3 = 20√3. Then R = (a√3) / 3 = (20√3 * √3) / 3 = (20 * 3) / 3 = 20. This matches the direct relation R = 2r, confirming the answer.
Why Other Options Are Wrong:
The value 5 cm is half the inradius and has no basis in the geometry of an equilateral triangle. The values 10√3 and 20√3 correspond to misinterpreting the formulas or mixing up side length with radius. The value 30 cm would imply R = 3r, which is not the correct relationship. Only 20 cm satisfies the simple and exact relation R = 2r for equilateral triangles.
Common Pitfalls:
Many learners forget the specific formulas for r and R in an equilateral triangle and try to use general formulas for other triangles, which are more complicated. Remembering that R is always exactly twice r in this special case is a useful shortcut that can save time in exams.
Final Answer:
The circumradius of the equilateral triangle is 20 centimetres.
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