Question

ABC is a right angled triangle with $\angle$ABC=${90}^{\circ }$, the centre of the circle passing through A,B,C 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

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 the center 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 ${180}^{\circ }$ since center 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 center cannot be on AB.

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

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

Therefore center lies on AC.