In the diagram, $A E$ and $B D$ are straight lines that intersect at $C$. If $B D=16, A B=9, C E=5$, and $D E=3$, then the length of $A C$ is
(A) 11
(B) 12
(C) 15
(D) 17
(E) 16

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-052.jpg?height=274&width=441&top_left_y=1077&top_left_x=1292)
In $\triangle C D E, C E=5, D E=3$, and $\angle C D E=90^{\circ}$.

By the Pythagorean Theorem, $C E^{2}=C D^{2}+D E^{2}$ or $C D^{2}=C E^{2}-D E^{2}$ or

$C D^{2}=5^{2}-3^{2}=25-9=16$, so $C D=4$ (since $C D>0$ ).

In $\triangle A B C, A B=9, B C=B D-C D=16-4=12$, and $\angle A B C=90^{\circ}$.

By the Pythagorean Theorem, $A C^{2}=A B^{2}+B C^{2}$ or $A C^{2}=9^{2}+12^{2}=81+144=225$, so $A C=15$ (since $A C>0)$.

ANsWeR: (C)