The Correct Answer and Explanation is:
The correct answer is A. incenter.
The term for the center of a circle inscribed within a triangle is the incenter. This is a specific point of concurrency within a triangle that has unique geometric properties.
An inscribed circle, also known as the incircle, is the largest circle that can be contained within a triangle. It is tangent to, or just touches, all three of the triangle’s sides without crossing any of them. The center of this incircle is the incenter. A key property of the incenter is that it is equidistant from all three sides of the triangle. This common distance is precisely the radius of the inscribed circle.
The incenter can be located by constructing the angle bisectors of the triangle. An angle bisector is a line or ray that divides an angle into two smaller, equal angles. When the three angle bisectors of a triangle are drawn, they all intersect at a single point. This point of intersection is the incenter.
It is important to distinguish the incenter from the other options. The circumcenter (Option B) is the center of the circumscribed circle, or circumcircle, which is the circle that passes through all three vertices of the triangle. The circumcenter is found at the intersection of the perpendicular bisectors of the triangle’s sides. A diameter (Option C) is not a center point; it is a line segment that passes through the center of a circle and connects two points on the circle’s edge. Therefore, the only correct term for the center of an inscribed circle is the incenter.
