There are eight equidistant points on a circle. How many right-angled triangle can be drawn using these points as vertices and taking the diameter as one side of the triangle?

Correct Option (A)

A right-angled triangle inscribed in a circle must have its hypotenuse as a diameter of the circle. This is a direct application of Thales's Theorem.

• There are 8 equidistant points on the circle. These points form 8/2 = 4 distinct diameters. For example, if the points are P1, P2, ..., P8, the diameters would be (P1, P5), (P2, P6), (P3, P7), and (P4, P8).
• For each diameter, any of the remaining (8 - 2) = 6 points on the circle can serve as the third vertex to form a right-angled triangle with the two points forming that diameter.
• Therefore, the total number of right-angled triangles is the number of diameters multiplied by the number of possible third vertices per diameter: 4 diameters × 6 points/diameter = 24 triangles.
• Alternatively, using the formula `2 * n * (n - 1)` where `n` represents the number of distinct diameters, and `n = 8/2 = 4` in this case: `2 * 4 * (4 - 1) = 2 * 4 * 3 = 24`.

Incorrect Options:

Options 16, 12, and 8 are incorrect because they do not align with the calculated number of right-angled triangles formed under the specified conditions, which is 24. These values would result from misapplication of the geometric principle or incorrect counting of diameters or vertices.