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. each vertex of the triangle lies outside the circle. c. the triangle is circumscribed about the circle. d. the circle is tangent to each side of the triangle. e. the circle is congruent to the triangle.

Explanation:

Step1: Recall definition of inscribed circle

An inscribed circle in a triangle is a circle that is drawn inside a triangle such that it touches each side of the triangle at exactly one - point.

Step2: Analyze vertex - circle position

The vertices of the triangle are outside the inscribed circle. If a vertex were inside the circle, the circle could not be tangent to all sides of the triangle. So, each vertex of the triangle lies outside the circle.

Step3: Understand circum - scription relationship

When a circle is inscribed in a triangle, the triangle is circumscribed about the circle. That is, the triangle surrounds the circle and the circle is tangent to each side of the triangle.

Step4: Analyze tangency

By the definition of an inscribed circle, the circle is tangent to each side of the triangle.

Step5: Analyze congruence

A circle and a triangle are different geometric shapes and cannot be congruent.

Answer:

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