Question

question 7 of 10

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

a. the circumcenter of a triangle is not always inside it.
b. the circumcenter is equidistant from each vertex of the triangle.
c. the circumcenter is at the intersection of the perpendicular bisectors of the triangles sides.
d. the circumcenter is equidistant from each side of the triangle.

Explanation:

Define the circumcenter

Using the Circumscribed Circle knowledge point, the circumcenter is the center of the circle that passes through all three vertices of a triangle.

Analyze the intersection property

The circumcenter is the point where the three perpendicular bisectors of the triangle's sides intersect. This makes statement C correct.

Analyze the distance to vertices

Since the circumcenter is the center of the circumscribed circle, the distance from this point to each vertex is equal to the radius of the circle. Thus, it is equidistant from all three vertices, making statement B correct.

Analyze the location property

The circumcenter lies inside an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle. Therefore, it is not always inside the triangle, making statement A correct.

Analyze the distance to sides

The point equidistant from each side of a triangle is the incenter, not the circumcenter. Thus, statement D is incorrect.

Answer:

• A. The circumcenter of a triangle is not always inside it. (Correct answer)
• B. The circumcenter is equidistant from each vertex of the triangle. (Correct answer)
• C. The circumcenter is at the intersection of the perpendicular bisectors of the triangle's sides. (Correct answer)
• D. The circumcenter is equidistant from each side of the triangle.