In an equilateral triangle, the radius of the circumcircle (the circle passing through all three vertices) is 14 cm. What is the radius, in centimetres, of the incircle (the circle inscribed inside the triangle and tangent to all three sides)?

Introduction:
This question assesses your understanding of special properties of an equilateral triangle, particularly the relationship between its circumradius (radius of the circumscribed circle) and inradius (radius of the inscribed circle). Knowing these standard geometric relationships allows you to solve the problem quickly without extensive trigonometry.

Given Data / Assumptions:

• The triangle is equilateral, meaning all sides and all angles are equal.
• Circumradius R = 14 cm.
• We must find the inradius r in centimetres.
• Standard geometric relations for equilateral triangles apply.

Concept / Approach:
For an equilateral triangle with side length a, the circumradius R and inradius r have the following relationship: R = 2r. This is a well known result derived from the geometry of equilateral triangles and the position of the centroid, incenter, and circumcenter, which all coincide in such a triangle. Therefore, if we know R, we can get r directly as r = R / 2.

Step-by-Step Solution:
Step 1: Use the relation between circumradius and inradius for an equilateral triangle: R = 2r. Step 2: The given circumradius is R = 14 cm. Step 3: Rearrange the formula to find r: r = R / 2. Step 4: Substitute R = 14: r = 14 / 2 = 7 cm.

Verification / Alternative check:
Another approach is to express R and r in terms of the side a. For an equilateral triangle: R = a / (sqrt(3)) and r = a * sqrt(3) / 6. Dividing R by r gives: R / r = [a / sqrt(3)] / [a * sqrt(3) / 6] = 6 / 3 = 2, so R = 2r again. With R = 14, we get r = 7, which confirms our earlier result.

Why Other Options Are Wrong:

• 14: This would imply r = R, which contradicts the known relation R = 2r.
• 28 and 35: These values are even larger than R, which is impossible because the incircle must fit inside the triangle.
• 21: Also larger than R, so it cannot represent the inradius.

Common Pitfalls:
Some learners confuse formulas of inradius and circumradius or misremember the relation. Others attempt to derive everything from scratch using height and side relationships, which is more time consuming. Memorising and correctly applying R = 2r for equilateral triangles is very helpful in competitive exams.

Final Answer:
The radius of the inscribed circle is 7 cm.