The probability that three points chosen at random inside a circle form an acute triangle is 1/3.
To determine the probability that three points chosen at random inside a circle form an acute triangle, consider the following approach:
Uniform distribution of points:
Assume that the points are selected uniformly at random within the circle. This means that each point has an equal chance of being chosen anywhere within the circle's area.
Acute triangle conditions:
An acute triangle is one in which all three angles are less than 90 degrees.
In the context of a circle, this means that the third point should lie within a specific region relative to the first two points to form an acute triangle.
Identifying the favorable region:
The favorable region for forming an acute triangle is the area inside the circle but outside the two isosceles right triangles formed by connecting the first two points and the center of the circle.
This region can be visualized as the area between the two arcs that subtend 120 degrees each.
Calculating the favorable area:
The favorable area can be determined using the formula for the area of a sector of a circle:
A_sector = (θ/360) * πr^2,
where θ is the central angle and r is the circle's radius. In this case, the favorable region corresponds to two sectors with central angles of 120 degrees each.
Therefore, the total favorable area is:
A_favorable = 2 * (120/360) * πr^2 = πr^2/3
Calculating the total area: The total area of the circle is πr^2.
Calculating the probability: The probability of selecting three points that form an acute triangle is equal to the ratio of the favorable area to the total area:
P(acute triangle) = A_favorable / A_total = (πr^2/3) / πr^2 = 1/3
Therefore, the probability that three points chosen at random inside a circle form an acute triangle is 1/3.