Question

question 2 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 inscribed in a given circle. d. many circles can be inscribed in a given triangle.

Explanation:

1. For option A: A circle can have infinitely many inscribed triangles. We can choose different sets of three points on the circumference of the circle to form different triangles. So, option A is incorrect.
2. For option B: The incircle of a triangle is unique. The in - center (center of the inscribed circle) is the point of intersection of the angle bisectors of the triangle, and there is exactly one such circle that is tangent to all three sides of the triangle. So, option B is correct.
3. For option C: Since we can pick any three non - collinear points on the circumference of a circle, many different triangular shapes can be formed. So, option C is correct.
4. For option D: There is only one incircle for a given triangle. The in - center and the radius of the incircle are uniquely determined by the triangle's side lengths and angles. So, option D is incorrect.

Answer:

B. Exactly one circle can be inscribed in a given triangle.
C. Many triangular shapes can be inscribed in a given circle.