Question

question 31 of 44 which of the following is a characteristic of the circumcenter of a triangle? a. it is equidistant from all three sides of the triangle. b. it is the center of the only circle that can be circumscribed about the triangle. c. it is the center of the circle that can be inscribed in a given triangle. d. it is the center of the only circle that can be inscribed in the triangle.

Explanation:

• Recall the definition of the circumcenter of a triangle: The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect. It is the center of the circumscribed circle (circumcircle) around the triangle.
• Analyze Option A: A point equidistant from all three sides of a triangle is the incenter (where angle bisectors meet), not the circumcenter. So A is incorrect.
• Analyze Option B: The circumcenter is indeed the center of the unique circle (circumcircle) that can be circumscribed about the triangle (passing through all three vertices). This matches the definition.
• Analyze Option C: The center of the circle inscribed in a triangle is the incenter, not the circumcenter. So C is incorrect.
• Analyze Option D: The center of the inscribed circle (incenter) is not the circumcenter. Also, the wording about "only circle that can be inscribed" is about the inradius circle, not related to circumcenter. So D is incorrect.

Answer:

B. It is the center of the only circle that can be circumscribed about the triangle.