In an equilateral triangle, each median has length 12 cm. Using standard formulas for equilateral triangles, find the inradius (radius of the incircle) of the triangle in centimetres.

Introduction / Context:

This geometry problem involves several special properties of an equilateral triangle. In an equilateral triangle, the medians, altitudes, angle bisectors, and perpendicular bisectors all coincide. Given the length of each median, you are asked to find the inradius, which is the radius of the circle that just touches all three sides from the inside. This requires connecting side length, altitude, and inradius using well known formulas.

Given Data / Assumptions:

• The triangle is equilateral.
• Each median has length 12 cm.
• In an equilateral triangle, the median equals the altitude.
• We must find the inradius r in centimetres.
• All sides are positive real lengths.

Concept / Approach:

For an equilateral triangle with side length a, the altitude h is given by h = (√3 / 2) * a. In an equilateral triangle, the altitude is also a median, so the given median length is the altitude. The inradius r is related to the side length by r = (a√3) / 6. We first use the altitude formula to find a, and then substitute into the inradius formula to obtain r.

Step-by-Step Solution:

Verification / Alternative check:

As a check, note that a = 24 / √3 can be written as 8√3 after rationalisation. Using a = 8√3, r = (8√3 * √3) / 6 = (8 * 3) / 6 = 24 / 6 = 4 again. Both forms give the same inradius, confirming the correctness of the calculation.

Why Other Options Are Wrong:

The values 3, 5, 6, and 8 would imply different relationships between the median and side length. Substituting these implied side lengths back into the altitude and inradius formulas would not keep the median at 12 cm. Only r = 4 cm is consistent with both the given median and the standard equilateral triangle formulas.

Common Pitfalls:

Some learners confuse the formulas for inradius and circumradius or forget that the median equals the altitude only in an equilateral triangle. Others incorrectly use h = a / 2 instead of h = (√3 / 2) * a. Remembering the exact forms of these special triangle formulas is crucial to avoid such errors.

Final Answer:

The inradius of the equilateral triangle with median 12 cm is 4 cm.