Question

the incenter of a triangle is shown below. which of the following statements is true?
point c is the point of concurrency for the medians of triangle wxy.
point c is equidistant from the sides of triangle wxy.
point c is the point of concurrency for the altitudes of triangle wxy
point c is equidistant from the vertices of triangle wxy.

Explanation:

The incenter of a triangle is the point where the angle bisectors intersect. By definition, the incenter is equidistant from all the sides of the triangle. Let's analyze each option:

• The point of concurrency for the medians is the centroid, not the incenter, so the first option is wrong.
• The incenter (point C here, as it's the center of the incircle) is equidistant from the sides, so this option is correct.
• The point of concurrency for the altitudes is the orthocenter, not the incenter, so the third option is wrong.
• The point equidistant from the vertices is the circumcenter, not the incenter, so the fourth option is wrong.

Answer:

Point C is equidistant from the sides of triangle WXY.