Final answer:
Using the secant segments power theorem, the equation (25)(42) = (2x)(x) is solved to find the length of segment PB. However, the resulting value does not match the provided choices, suggesting an issue with the question data or choices.
Step-by-step explanation:
To solve for the length of the segment PB using the given values of PA, CA, and the condition PD = DB, we can apply the secant segments power theorem. This theorem states that if two secant segments are drawn to a circle from an external point, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
In formula terms, this is expressed as (PA)(PC) = (PB)(PD). We are given that PA = 25 and CA = 17 which means PC = PA + CA = 25 + 17 = 42. We are also given that PD = DB; since PB = PD + DB, we can let x equal the length of PD (and DB), so PB = 2x.
Substitute the known values into the equation gives us (25)(42) = (2x)(x). Simplify to get 1050 = 2x^2, and then solving for x we get x^2 = 525. Taking the square root of both sides, x = √525, which simplifies to x = 22.9 (approx). Therefore, PB = 2x is approximately 45.8, which is not among the provided choices, indicating a potential issue with the original values or the question choices.