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

ABC is a triangle. So points A, B, C are noncollinear.

$\therefore$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 =${90}^{\circ }$ is not possible, since sum of internal angles in a triangle is ${180}^{\circ }$,

and it is already given that $\angle$ABC=${90}^{\circ }$.

$\Rightarrow$ 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.