Show that the triangle formed by a chord and the centre of the circle is isosceles.
Let AB be a chord of a circle with centre C. We join the centre C to the endpoints A and B of the chord, forming triangle CAB. In this triangle, the sides CA and CB represent the radius of the circle.
Since the radii of a circle are equal in length, we have CA = CB. A triangle that has two equal sides is defined as an isosceles triangle. In triangle CAB, the base is AB, and the two equal sides are CA and CB.
Because CA = CB, triangle CAB is isosceles. Hence, it is proven that the triangle formed by a chord and the centre of the circle is an isosceles triangle.
Explanation
The textbook context explains in Theorem 3 that for a chord AB and centre C, AC = BC = the radius of the circle. Since a triangle with two equal sides is isosceles, triangle CAB must be an isosceles triangle.
Solution Steps
Step 1: Draw a chord AB and join its endpoints to the centre C, forming triangle CAB.
Step 2: Note that CA and CB are radii of the circle, so CA = CB.
Step 3: Conclude that triangle CAB is isosceles because it has two equal sides (CA = CB).