Question

the center of the circumscribed circle of the triangle lies on segment __, which is __ of the circle.

Answer 1

"the center of the circumscribed circle of the triangle lies on segment AC, which is diameter of the circle."

A proper and general statement would be:

"The center of the circumscribed circle of the triangle lies on the perpendicular bisectors of its sides, which is equidistant from all three vertices of the triangle."

Perpendicular Bisectors:

The perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to it. For each side of the triangle, there is a corresponding perpendicular bisector.

Circumscribed Circle (Circumcircle):

The circumscribed circle of a triangle is a circle that passes through all three vertices of the triangle. The center of this circle is called the circumcenter.

In the case of right triangles, the hypotenuse is the diameter of the circle.

Equidistant from Vertices:

The circumcenter is equidistant (at the same distance) from all three vertices of the triangle. This means that the distances from the circumcenter to each vertex are equal.

Answer 2

Given the right triangle:

ABC

Where angle B is the right angle.

Let's complete the given statement.

To find where the center of the circumscribed cirlce lies, let's first sketch the triangle and the circle.

A circumscribed circle is a circle which passes through all the vertices of a triangle.

By applying the converse of Thales theorem, we can see that the point D is center of the cirlce.

The point lies on line segment AC.

Also, the diameter of a circle is a line which passes through the center of the circle and touch the circumference at both ends.

Hence, line AC is the diameter of the circle.

Therefore, the complete statement is:

The center of the circumscribed circle of the triangle lies on segment segment AC, which is the diameter of the circle.

ANSWER:

The center of the circumscribed circle of the triangle lies on segment segment AC, which is the diameter of the circle.