AMC 12 2025a Problem 1

Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1{:}30$, traveling due north at a steady $8$ miles per hour. Betsy leaves on her bicycle from the same point at $2{:}30$, traveling due east at a steady $12$ miles per hour. At what time will they be exactly the same distance from their common starting point?

$\textbf{(A) } 3{:}30 \textbf{(B) } 3{:}45 \textbf{(C) } 4{:}00 \textbf{(D) } 4{:}15 \textbf{(E) } 4{:}30$