Question

question 3 of 10
what can be said about the relationship between triangles and circles? check all that apply.
a. exactly one circle can be inscribed in a given triangle.
b. many triangular shapes can be circumscribed about a given circle.
c. many circles can be inscribed in a given triangle.
d. exactly one triangle can be inscribed in a given circle.

Explanation:

• Option A: Every triangle has exactly one incircle, tangent to all three sides, determined by the intersection of angle bisectors.
• Option B: A given circle can have infinitely many different triangles circumscribed around it (triangles where the circle is the incircle), varying in side lengths and angles.
• Option C: A triangle only has one unique incircle, so this is false.
• Option D: A given circle can have infinitely many inscribed triangles (with vertices on the circle), so this is false.

Answer:

A. Exactly one circle can be inscribed in a given triangle.
B. Many triangular shapes can be circumscribed about a given circle.