Final answer:
The perimeter of a circle inscribed in a pentagon is 5 times the radius of the inscribed circle (option A).
Step-by-step explanation:
The perimeter of a circle inscribed in a pentagon is 5 times the radius of the inscribed circle (option A). To understand why, we can first calculate the perimeter of the pentagon using the formula P = 5s, where s is the length of each side of the pentagon. By drawing lines from the center of the circle to each vertex of the pentagon, we create 5 congruent triangles. These triangles are isosceles, meaning they have two sides of equal length.
The length of one side of the pentagon is equal to the sum of the radius of the inscribed circle and the base of the isosceles triangle. Using the Pythagorean theorem, we can find the base of the triangle as 2 times the radius. Therefore, s = r + 2r = 3r. Substituting this value into the perimeter formula, we get P = 5(3r) = 15r, which is 5 times the radius of the inscribed circle.