Question

8. Point O is the circumcenter of the triangle ABC shown below.

Which segment passes through point O for all lengths of sides of the triangle?
A. angle bisector of angle ABC
B. perpendicular bisector of side AB
C. a line segment drawn from vertex C to bisect side AB
D. a line segment drawn from vertex A to cut side BC at right angles

Answer 1

The circumcenter O, formed by the intersection of the perpendicular bisectors of sides AB and BC, is equidistant from all vertices, with OA = OB = OC. Here option B is correct.

In a triangle, the circumcenter is the point where the perpendicular bisectors of its sides intersect. In this case, the circumcenter O is formed by the intersection of the perpendicular bisector of side AB and the perpendicular bisector of side BC.

The circumcenter is equidistant from all three vertices, making OA, OB, and OC equal, and these distances represent the radius of the circumcircle.

This line not only bisects side AB but also intersects with the perpendicular bisector of side BC at the circumcenter O. The equality of OA, OB, and OC ensures that the circumcircle passes through all three vertices of the triangle, making it a significant point in the context of the triangle's geometry. Here option B is correct.

Answer 2

Option B. perpendicular bisector of side AB

Explanation:

we know that

The circumcenter is the intersection point of the perpendicular bisectors of a triangle

so

In this problem

The circumcenter O is the intersection point of perpendicular bisector of side AB and perpendicular bisector of side BC

OA=OB=OC ------> the radius of the circle that passes through all three of the triangle's vertices.

therefore

Option B. perpendicular bisector of side AB