Question

question 4 of 10

what can be said about the relationship between triangles and circles? check all that apply.

a. exactly one triangle can be inscribed in a given circle.
b. exactly one circle can be inscribed in a given triangle.
c. many triangular shapes can be circumscribed about a given circle.
d. many circles can be inscribed in a given triangle.

Explanation:

Analyze inscribed triangles in a circle

Using the Inscribed Triangle and Circumscribed Circle knowledge points, we analyze option A. For any given circle, we can choose any three distinct points on its circumference to form an inscribed triangle. Since there are infinitely many ways to choose these three points, infinitely many different triangles can be inscribed in a single given circle. Therefore, option A is incorrect.

Analyze inscribed circles in a triangle

Using the Inscribed Circle knowledge point, we analyze option B and option D. Every triangle has a unique point called the incenter, which is the intersection of its angle bisectors. The incenter is equidistant from all three sides of the triangle. This distance is the inradius. Thus, there exists exactly one unique circle tangent to all three sides of a given triangle. Therefore, option B is correct, and option D is incorrect.

Analyze circumscribed triangles about a circle

Using the Circumscribed Circle and Inscribed Circle knowledge points, we analyze option C. A triangle is circumscribed about a circle if the circle is inscribed in the triangle (tangent to all three sides). We can construct many different triangles that are tangent to a single given circle by choosing different sets of three tangent lines. Therefore, infinitely many triangular shapes can be circumscribed about a given circle. This makes option C correct.

Answer:

• (A) Exactly one triangle can be inscribed in a given circle.
• (B) Exactly one circle can be inscribed in a given triangle. (Correct answer)
• (C) Many triangular shapes can be circumscribed about a given circle. (Correct answer)
• (D) Many circles can be inscribed in a given triangle.