Point o is the center of a circle passing through points a, b, and c. ∠b is a right angle.
the center of the circumscribed circle lies on line segment, (options: ab, bc, or ca) and the longest side of the triangle is equal to the (options: radius, diameter, or circumference) of the circle.

 The answers are CA and diameter.

Step-by-step explanation:  As given in the question, a circle with centre 'O' is passing through three A, B and C, where ∠B is a right angle.

In the circumscribed circle, we have the converse of the Thales' theorem, which states that -

if A, B, and C are distinct points on a circle where angle ∠ABC is a right-angle, then the line AC will be a diameter of the circle.

The given information is drawn in the attached figure.

Applying the converse of Thales' Theorem, we have AC will be a diameter of the circle.

We know that the diameter of a circle is a chord parring through the centre, so the centre will lie on the line segment CA.

Since the opposite side of the right-angle is the longest side (hypotenuse) in a right-angled triangle, so we have CA as the longest side of ΔABC.

Since CA is also a diameter of the circle, so the longest side of the triangle is equal to the diameter of the circle.

Thus, the centre will lie on the line segment CA and the longest side of the triangle is equal to the diameter of the circle.

CA and Diameter

Step-by-step explanation: