Question

if a circle is inscribed in a triangle, which of the following must be true? check all that apply. a. each vertex of the triangle lies inside the circle. b. the circle is congruent to the triangle. c. the triangle is circumscribed about the circle. d. the circle is tangent to each side of the triangle. e. each vertex of the triangle lies outside the circle.

Explanation:

• For option A: The vertices of a triangle with an inscribed circle are outside the circle, not inside, so this is false.
• For option B: A circle and a triangle are different geometric shapes and cannot be congruent, so this is false.
• For option C: When a circle is inscribed in a triangle, the triangle is defined as being circumscribed about the circle, so this is true.
• For option D: By definition, an inscribed circle in a triangle touches each side of the triangle at exactly one point, meaning it is tangent to each side, so this is true.
• For option E: All vertices of the triangle are outside the inscribed circle, as the circle is contained within the triangle, so this is true.

Answer:

C. The triangle is circumscribed about the circle.
D. The circle is tangent to each side of the triangle.
E. Each vertex of the triangle lies outside the circle.