Question

What can be said about the relationship between triangles and circles? Choose all that apply.

Answer 1

Answer:  The answers are (B) and (D).

Step-by-step explanation:  We are given four options describing the relationship between circles and triangles. We are to select the correct statements.

Since a circle has infinitely many non-collinear points, and we need three non-collinear points to draw a triangle, so we can draw many triangular shapes can be inscribed in a given circle.

So, option (A) is incorrect and (B) is correct.

Now, we can draw a circle with centre as the in centre of a triangle and a triangle has only one in centre, so we can inscribe many circles inside a triangle with centre as incentre.

Hence, option (C) is incorrect.

Also, considering the construction symmetry, we can conclude that exactly one circle can be circumscribed about a given triangle. Therefore, option (D) is also correct.

Thus, (B), (C) and (D) are correct.

Answer 2

The correct answers are:

B) many triangular shapes can be inscribed in a given circle; and
D) exactly one circle can be circumscribed about a given triangle.

Explanation:

One of our geometric theorem states that for any 3 non-collinear points, exactly one circle can be drawn through it. This explains answer D; the non-collinear points would be the vertices of the triangle, and only one circle can be drawn through those points (circumscrbied about the triangle).

For B, we can have many triangles inscribed in a circle; the differences in the measures of the angles used will create the different triangles.