Geometry

Question Analysis

This question focuses on determining which point (or points) among the given options is equidistant from the sides of a triangle, testing knowledge of triangle centers.

Key Concept Explanation

Circumcenter: The point of intersection of the perpendicular bisectors of a triangle's sides. It is equidistant from the vertices of the triangle, not from the sides.

Incenter: The point of intersection of the angle bisectors of a triangle. It is equidistant from the sides of the triangle.

Step - by - Step Solution

1. Recall the properties of the circumcenter:

The circumcenter is defined as the point where the perpendicular bisectors of the triangle's sides meet. Its key property is that it is equidistant from the vertices of the triangle. Since it has no direct relation to being equidistant from the sides, it does not meet the requirement.

2. Recall the properties of the incenter:

The incenter is the intersection of the angle bisectors of a triangle. By the property of angle bisectors (any point on an angle bisector is equidistant from the two sides of the angle), the incenter is equidistant from all the sides of the triangle.

Option Analysis

A.Circumcenter: Incorrect. As it is equidistant from the vertices, not the sides.

B.Incenter: Correct. Because it is the intersection of angle bise...