Question

if a circle is inscribed in a triangle, which of the following must be true? check all that apply.

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

Explanation:

Analyze the definition of an inscribed circle

An inscribed circle (or incircle) of a triangle is the largest circle contained entirely within the triangle; it is tangent to all three sides of the triangle.

Evaluate each statement based on the definition

• A: A circle and a triangle are different shapes and cannot be congruent. (False)
• B: By definition, an inscribed circle is tangent to each side of the triangle. (True)
• C: Since the circle is inside the triangle, the vertices of the triangle must lie outside the circle, not inside. (False)
• D: If a circle is inscribed in a triangle, then the triangle is circumscribed about the circle. (True)
• E: Since the circle is entirely inside the triangle, all three vertices of the triangle lie outside the circle. (True)

Select the correct options

The true statements are B, D, and E.

Answer:

• A. The circle is congruent to the triangle.
• B. The circle is tangent to each side of the triangle. (Correct answer)
• C. Each vertex of the triangle lies inside the circle.
• D. The triangle is circumscribed about the circle. (Correct answer)
• E. Each vertex of the triangle lies outside the circle. (Correct answer)