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

a. 24

b. 16

c. 12

d. 8

Correct Answer: a. 24

#### Question from UPSC Prelims 2022 CSAT Paper

**Explanation : **

## There are eight equidistant points on a circle

**Angle subtended by the diameter chord at the circumference of the circle is 90.**

To form a right-angled triangle using the diameter of the circle as one side and the eight equidistant points as vertices, we need to select two points on the circle that are equidistant from both ends of the diameter.

There are four such pairs of points.

To form a right-angled triangle, we need to choose one of these pairs of points as the endpoints of the diameter and then select one of the remaining six points as the third vertex. There are six ways to choose the third vertex for each pair of points, so the total number of right-angled triangles is: 4*6 = 24