Question

ABC is the right angled triangle with $\angle$ABC $={90}^{\circ }$, The centre of the circle passing through ABC lies on _______.

• A. AB
• B. AC
• C. BC
• D. a circle cannot be drawn through passing through A,B,C

Answer 1

The correct option is B

AC

Since, ABC is a triangle, so points A, B, C are non-collinear. Therefore, a circle can be drawn passing through these three points. Now, assume that the centre is on AB.

We know that angle subtended by a chord at the center is twice the angle subtended by it at any point on the circle.

The angle subtended by AB at the center is ${180}^{\circ }$ as the centre lies on AB.

Therefore, $\angle$ACB $={90}^{\circ }$ which is not possible since sum of internal angles in a triangle is ${180}^{\circ }$ and it is already given that $\angle$ABC $={90}^{\circ }$

So, centre cannot lie on AB.

Similarly, you can prove that centre does not lie on BC.

If we assume that centre lies on AC, then $\angle$ABC should be ${90}^{\circ }$ which is given.

Therefore, centre lies on AC.