With 3 points selected randomly on a circle, you may draw infinite number of triangles. However, the condition here is that we have to count only acute triangles. So, basically the question is how many types of triangles we can draw with these 3 points. Based on the measure of angles in triangles, 3 types of triangles can drawn - acute, obtuse, and right angled triangle. The other types will fall into one of these types. For instance, equilateral triangle will be an acute triangle after all. Let's figure out how many cases of these types of triangles can be drawn to find out what proportion of acute triangle exists.
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In the figure, I selected points A and B randomly, drawing a chord AB. Then I focused on the larger arc of chord AB and selected C randomly on the larger arc. I named these several points for C as P, Q, R, and S. This way several types of triangles can be drawn with chord AB and each of the other random points P, Q, R, and S.
Starting from point any point A or B, 3 types of triangles can be formed - acute, obtuse, and right angle triangle. As shown in the figure, point A may form an obtuse triangle APB, a right angle triangle AQB, or an acute angle triangle ASB. However, one more right angle triangle ABR may be drawn with right angle at point B.
Therefore, this way each point A, B, and C has four possibilities - acute, obtuse, two right angle triangles. Probability of acute angle will one in four possible types of triangle. Hence, correct answer is C.