Final answer:
The possible arc lengths for the concentric circles, formed by the same central angle, are those that maintain a 3:5 ratio, in direct proportion to the radii of the circles. Hence, the correct answers are C. 3 and 5 and D. 45 and 75.
Step-by-step explanation:
The question asks us to determine the possible arc lengths for two concentric circles with radii of 15 and 25 respectively, given that the arcs are formed by the same central angle. To find the arc lengths, we need to understand that the arc length is directly proportional to the radius of the circle. Thus, the ratio between the arc lengths of the two circles will be the same as the ratio between their radii. For circles with radii of 15 and 25, the ratio is 15:25, or 3:5 when simplified.
Since the arc length is proportional to the radius, a central angle that subtends an arc length of 'x' on the smaller circle will subtend an arc length of \((5/3) * x\) on the larger circle. Therefore, the possible pairs of arc lengths would be in a ratio of 3:5, which is the case for the pair C. 3 and 5 as well as the pair D. 45 and 75. These are the only two pairs that maintain the proportional relationship between the arc lengths and the radii of the two concentric circles.