Ian Stewart wrote:Using only Statement 2, you can be certain that BC is a diameter. You do not, however, have any information about the location of point A. It might be that point A is extremely close on the circle to point B, for example, in which case the area of the triangle will be very small, or A could be halfway between B and C, and the area could be as you've calculated. So we need Statement 1.

I think you've assumed that AO, where O is the center of the circle, will be perpendicular to BC. That's not usually going to be true - in fact that's only true if AB = AC, which is why we need Statement 1.

Ian Stewart wrote:

johnconnor99 wrote: Hi Ian,

Thanks a lot for the quick reply!! Yeah..That was the assumption I made because of the fact that, the chords emanating from a diameter always subtends a rtangle and LA is that right angle. Please correct me if I'm wrong.

Yes, if BC is a diameter, then ABC is a right triangle here, with the right angle at A. But if you take BC as the base of your triangle, then you don't really care if angle A is a right angle. Instead, you need a right angle where you connect your height to BC. I think you've assumed that OA, where O is the center of the circle, is at 90 degrees to BC, and that is almost never going to be true (if you draw points B and A very close together, you can see that); OA will only be perpendicular to BC if the triangle is isosceles. If the triangle is not isosceles, then the point on BC that you would join to A to make a height will not be the center of the circle.

OA is of course a radius, so its length is 6. If you can be certain that OA is perpendicular to BC, then OA is the height of the triangle, and the area is indeed 6*12/2 = 36, but you need both statements to be sure that OA is a height.