### Instruction: In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 \qquad \textbf{(E)}\ 7:1$.### Response: I want to find the ratio of $PA$ to $AB$, so I need to compare the lengths of those two segments.
I notice that $P$ is the point where the external angle bisector of $C$ meets the extended side $BA$, so I recall a property of external angle bisectors: they divide the opposite side externally in the ratio of the adjacent sides.
That means that $PA:AB = AC:CB$, which is given as $3:4$.
Therefore, the answer is (B) $3:4$.