M17-16

Official Solution:


Which of the following can be the perimeter of a triangle inscribed in a circle with a radius of 1?

I. 0.001

II. 0.010

III. 0.100


A. I only
B. III only
C. II and III only
D. I, II, and III
E. None of the above.


First of all, note that any three points on any circle form a triangle. If we imagine three points on a circle and move them closer to each other, we can decrease the perimeter of the triangle they form. Since we can position the points as close together as we want, we can make the perimeter of the triangle arbitrarily small, approaching zero. Thus, we can inscribe a triangle in any circle of any radius such that its perimeter is as small as desired. This implies that the lower limit of the perimeter of a triangle inscribed in any circle of any radius is zero, making answer D correct.

Furthermore, among all triangles inscribed in a given circle, the equilateral triangle has the largest perimeter. An equilateral triangle inscribed in a circle of radius 1 has a perimeter of \(3\sqrt{3}\). Hence, the range of perimeters inscribed in a circle of radius 1 is \(0 < p \leq 3\sqrt{3}\).


Answer: D