If point $z$ is equidistant from the sides of triangle $RST$, which of the following must be true?

• A. Point $z$ is the centroid of triangle $RST$.
• B. Point $z$ is the incenter of triangle $RST$.
• C. Point $z$ lies on the angle bisector of one of the angles of $RST$.
• D. Point $z$ is the circumcenter of triangle $RST$.

Answer: (B)

Verified step by step guidance

1. Recall the definitions of special points in a triangle: the centroid is the intersection of medians, the circumcenter is the intersection of perpendicular bisectors of sides, and the incenter is the intersection of angle bisectors.
2. Understand that the incenter is the unique point inside a triangle that is equidistant from all three sides of the triangle.
3. Recognize that being equidistant from the sides means the point lies on the angle bisectors of the triangle, since the incenter is the point where all three angle bisectors meet.
4. Note that the centroid relates to medians and the circumcenter relates to perpendicular bisectors of sides, which do not guarantee equal distance from the sides.
5. Conclude that if a point is equidistant from the sides of a triangle, it must be the incenter, and it lies on the angle bisectors of the triangle.