Question

question 5 of 10

which of the following are properties of the incenter of a triangle? check all that apply.

a. the incenter is where all of the bisectors of the angles of the triangle meet.
b. the incenter is where all of the perpendicular bisectors of the sides intersect.
c. the incenter of a triangle is always inside it.
d. the incenter of a right triangle lies on the side opposite the right angle.

Explanation:

Define the incenter of a triangle

The incenter is the center of the Inscribed Circle (incircle) of a triangle. By definition, it is the point where the three internal angle bisectors of the triangle intersect. This makes statement A correct.

Determine the location of the incenter

Since the incenter is the center of a circle entirely contained within the triangle, it must always lie inside the triangle. This is true for all triangles (acute, right, and obtuse). Thus, statement C is correct.

Analyze the incorrect options

Statement B describes the circumcenter, which is the intersection of the perpendicular bisectors of the sides. Statement D describes a property of the circumcenter of a right triangle (which lies on the hypotenuse), not the incenter.

Summarize the correct properties

The correct properties of the incenter among the choices are that it is the intersection of the angle bisectors (A) and it always lies inside the triangle (C).

Answer:

• A. The incenter is where all of the bisectors of the angles of the triangle meet. (Correct answer)
• B. The incenter is where all of the perpendicular bisectors of the sides intersect.
• C. The incenter of a triangle is always inside it. (Correct answer)
• D. The incenter of a right triangle lies on the side opposite the right angle.