2024 TMJC H1 Q7

Eight points arranged on a circle

Eight points, labelled \(A\) to \(H\), are arranged equally spaced in a circle as shown in the diagram. A triangle can be formed by connecting three points with three line segments. For example, triangle \(ABC\) can be formed by connecting points \(A\), \(B\) and \(C\) with line segments \(AB\), \(BC\) and \(AC\).

1. Find the number of possible triangles that can be formed using the eight points.[1]
2. How many of the triangles formed using the eight points are right-angled triangles? You may use the property that an angle inscribed in a semicircle is a right angle. For example, triangle \(ABE\) is a right-angled triangle.[2]
3. 
   A point, \(J\), is added in the centre of the circle as shown in the diagram. Find the number of possible triangles that can be formed using the nine points. One possible triangle is triangle \(ABF\).