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

The correct option is 24.

To find the number of right-angled triangles that can be drawn using eight equidistant points on a circle, we can use Thales's theorem from geometry.
Thales's theorem states that if three points $A$, $B$, and $C$ lie on a circle where the line segment $AB$ is a diameter of the circle, then the angle $\angle ACB$ is a right angle (90 degrees).
Therefore, any right-angled triangle inscribed in a circle must have a diameter of the circle as its hypotenuse (one of its sides).

Let us break down the calculation step-by-step:

Step 1: Determine the number of diameters.
We are given 8 equidistant points on the circle. A diameter connects two opposite points. Since the points are equidistant, we can pair each point with its directly opposite point to form a diameter.
The number of available diameters is:
$\frac{8}{2}=4$
So, there are 4 unique diameters.

Step 2: Determine the number of right-angled triangles per diameter.
For any chosen diameter, we have used 2 of the 8 points as the endpoints of the diameter (the hypotenuse of our right-angled triangle).
To form a triangle, we need to choose a third vertex from the remaining points on the circle.
The number of remaining points is:
$8-2=6$
According to Thales's theorem, choosing any of these 6 remaining points as the third vertex will form a right-angled triangle with the chosen diameter as its hypotenuse. Thus, each diameter gives rise to exactly 6 right-angled triangles.

Step 3: Calculate the total number of right-angled triangles.
Since there are 4 diameters, and each diameter can form 6 right-angled triangles, the total number of right-angled triangles is:
$4\times 6=24$

Thus, we can draw a total of 24 right-angled triangles using the 8 points.