Question

in the figure below, two congruent (identical) circles with centers a and b touch only at point c. if \\( \overline{ab} \\) is 30 ft, what is the circumference of circle a? leave your answer in terms of \\( \pi \\).

Explanation:

Step1: Determine the radius of circle A

Since the two circles are congruent and touch at point C, the distance \( AB = 30\pi \) (wait, no, looking at the diagram, the length \( AB \) is the sum of the radii of the two circles. But since they are congruent, the radius of each circle is half of \( AB \). Wait, the diagram shows \( AB = 30 \) (maybe the \( \pi \) was a typo? Wait, the problem says "If \( AB \) is \( 30\pi \)" no, wait the user's image: the length \( AB \) is labeled 30? Wait, the problem: "two congruent circles with centers A and B touch only at point C. If \( AB = 30 \) (maybe the \( \pi \) is a typo, or maybe the length is 30 units). Wait, congruent circles have the same radius. The distance between centers A and B is the sum of their radii, but since they are congruent, radius \( r_A = r_B \), so \( AB = r_A + r_B = 2r_A \). So if \( AB = 30 \) (assuming the label is 30, not 30π), then \( r_A = \frac{30}{2} = 15 \). Wait, but the problem says "Leave your answer in terms of \( \pi \)". So circumference of a circle is \( C = 2\pi r \). So if radius is 15, then circumference is \( 2\pi \times 15 = 30\pi \). Wait, maybe the \( AB \) length is 30 (not 30π). Let's re-express:

Step1: Find radius of circle A

The two circles are congruent, so their radii are equal. The distance between centers A and B is \( AB \), which is the sum of the radii of circle A and circle B. Since \( r_A = r_B \), \( AB = r_A + r_B = 2r_A \). Given \( AB = 30 \) (assuming the diagram's label is 30, maybe the problem had a typo with \( \pi \)), so \( r_A = \frac{AB}{2} = \frac{30}{2} = 15 \).

Step2: Calculate circumference of circle A

Circumference formula: \( C = 2\pi r \). Substitute \( r = 15 \): \( C = 2\pi \times 15 = 30\pi \).

Answer:

\( 30\pi \)