Final answer:
PB is equal to PD because both are radii of the circumcircle of ∆ABC with P as the circumcenter. Since PD is given as 12, PB is also 12.
Step-by-step explanation:
If P is the circumcenter of ∆ABC, and D is a point on line segment AB, then by definition of the circumcenter, PD is equal to PB because they are both radii of the circumcircle of ∆ABC. Given AD = 8x + 3 and DB = 17x - 15, and we know that PD = 12 (a radius of the circumcircle), we can find the length of PB. The sum of segments AD and DB gives us AB, so:
AD + DB = AB
(8x + 3) + (17x - 15) = AB
(25x - 12) = AB
Since P is the circumcenter and D lies on AB, the length of PD (which is also the radius of the circumcircle) is equal to the length of PB. Therefore, PB also equals 12.
The student's question "What is PB?" is answered by recognizing that PB is equal to PD, which is given as 12. No need to solve for x as the length of PB is known from the circumcenter property. Thus, PB = 12.