Final Answer:
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 [BC], and the longest side of the triangle is equal to the [diameter] of the circle thus option D is correct.
Step-by-step explanation:
In a right-angled triangle with a circumscribed circle, the center of the circle lies on the hypotenuse. Therefore, the correct option for the line segment containing the center of the circumscribed circle is [BC]. Additionally, the longest side of the triangle, which is the hypotenuse [BC], is equal to the diameter of the circumscribed circle.
Understanding the relationship between the circumcircle and the right-angled triangle is essential. For any right-angled triangle, the center of the circumcircle lies on the midpoint of the hypotenuse. This is a consequence of the fact that the circumcircle is the unique circle that passes through all three vertices of the triangle. The diameter of the circumcircle is equal to the hypotenuse of the right-angled triangle.
In this specific case, selecting [BC] as the line segment containing the center and [diameter] as the term representing the longest side aligns with the general properties of a right-angled triangle and its circumscribed circle (option D). This understanding is crucial in geometry, especially when dealing with the relationships between triangles and circles, as it enables the correct identification of geometric elements and their properties.