Question

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

Explanation:

Step1: Define inscribed - circle relationship

When a circle is inscribed in a square, the circle touches each side of the square at exactly one point. This means the circle is tangent to each side of the square, so B is true.

Step2: Analyze vertex - circle position

The radius of the inscribed circle is less than half of the length of the diagonal of the square. So each vertex of the square lies outside the circle, and C is true.

Step3: Understand circumscribed - square concept

If a circle is inscribed in a square, then the square is circumscribed about the circle. That is, the square surrounds the circle and the circle is tangent to the square's sides, so E is true.

Step4: Analyze congruence

A circle and a square are different geometric shapes and cannot be congruent (congruent means same shape and size), so D is false.

Step5: Analyze vertex - inside position

Since the circle is inscribed and vertices are outside, A is false.

Answer:

B. The circle is tangent to each side of the square., C. Each vertex of the square lies outside the circle., E. The square is circumscribed about the circle.