Question

question 38 of 44
which of the following statements is true regarding the relationship between
circles and triangles?

a. there are many triangles that can be inscribed in a given circle.

b. there are many circles that can be circumscribed about a triangle.

c. there are many circles that can be inscribed in a given triangle.

d. there is only one unique circle that can be circumscribed about a
triangle.

Explanation:

• For option A: A given circle can have infinitely many inscribed triangles (by choosing different sets of three points on the circle), so this statement is false.
• For option B: A triangle has exactly one circumcircle (circumscribed circle) because the perpendicular bisectors of the triangle's sides intersect at a unique point (circumcenter) which is equidistant from all three vertices, so there's only one circle that can be circumscribed about a triangle, making this statement false.
• For option C: A triangle has exactly one incircle (inscribed circle) because the angle bisectors of the triangle intersect at a unique point (incenter) which is equidistant from all three sides, so there's only one circle that can be inscribed in a triangle, making this statement false.
• For option D: As explained, the circumcenter (intersection of perpendicular bisectors of a triangle's sides) is unique, and the circumradius (distance from circumcenter to any vertex) is unique, so there is only one unique circle that can be circumscribed about a triangle. This statement is true.

Answer:

D. There is only one unique circle that can be circumscribed about a triangle.