The perimeter of an equilateral triangle is 12√3 cm. Find the radius of the circle inscribed in the triangle.

We know from the condition that the perimeter of an equilateral triangle is 12√3.

Let’s start the solution by labeling the triangle as ABC.

Knowing from the condition that it is equilateral, we can find the lengths of the sides of the triangle.

AB = BC = AC = 12√3 / 3 = 4√3.

It is known that in an equilateral triangle the center of the incircle and the circumcircle coincide with the point of intersection of the medians.

And it is also known that in an equilateral triangle, all angles are equal to 60 °.

Let’s draw the height BK, it is also the median.

The resulting triangle ABK:

BK = AB * sin 60 ° = 4√3 * √3 / 2 = 6.

Using the median property of a triangle:

OK = BK / 3 = 6/3 = 2.

Answer: r = 2.