In the diagram, points $A, B$ and $C$ are plotted on a $7 \times 10$ grid. Line segments join $A, B$ and $C$. An ant walks directly from $A$ to $B$ to $C$ to $A$ along these line segments. The distance that the ant walks is equal to
(A) $40+\sqrt{13}$
(B) $13+\sqrt{39}$
(C) $15+\sqrt{149}$
(D) $13+\sqrt{89}$
(E) 26

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-003.jpg?height=306&width=401&top_left_y=232&top_left_x=1312)

## Part B: Each correct answer is worth 6.
The path that the ant walks from $A$ to $B$ is vertical and has length 5 .

The path that the ant walks from $B$ to $C$ is horizontal and has length 8 .

The path that the ant walks from $C$ to $A$ does not follow the gridlines. To determine the length of $C A$, we can use the Pythagorean Theorem because $A B$ and $B C$ meet at a right angle.

This gives $C A^{2}=A B^{2}+B C^{2}=5^{2}+8^{2}=25+64=89$.

Since $C A>0$, then $C A=\sqrt{89}$.

Thus, the total distance that the ant walks is $5+8+\sqrt{89}$ or $13+\sqrt{89}$.

ANSWER: (D)