Question

what characterizes the incircle of a triangle? a. it passes through all three vertices b. it is the same as the circumcircle c. it is inscribed within a triangle and touches all three sides d. it is equidistant from all vertices the angle bisectors of a triangle intersect at which point of concurrency? a. orthocenter b. incenter c. circumcenter d. centroid which statement is true about the circumcircle of a triangle? a. it is always inside the triangle b. it passes through all the vertices of a triangle c. it is tangent to all three sides d. its center is the centroid

Explanation:

1. The incircle of a triangle is a circle that is inscribed within the triangle and touches all three sides. A circle passing through all three vertices is the circum - circle. The center of the incircle is equidistant from the sides, not vertices.
2. The angle bisectors of a triangle intersect at the incenter. The orthocenter is the intersection of the altitudes, the circumcenter is the intersection of the perpendicular bisectors of the sides, and the centroid is the intersection of the medians.
3. The circum - circle of a triangle passes through all the vertices of the triangle. It is not always inside the triangle (it is outside for an obtuse - angled triangle), and it is not tangent to the sides (the incircle is tangent to the sides), and its center is the circumcenter, not the centroid.

Answer:

1. c. It is inscribed within a triangle and touches all three sides
2. b. Incenter
3. b. It passes through all the vertices of a triangle