1. **State the problem:** We have a right triangle with vertices Indianapolis, Dayton, and Lima. The side between Indianapolis and Dayton is 165, the hypotenuse between Indianapolis and Lima is 173, and the side between Dayton and Lima is $x$. We need to find $x$. 2. **Formula used:** In a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse. 3. **Apply the formula:** Here, $a = 165$, $b = x$, and $c = 173$. So, $$165^2 + x^2 = 173^2$$ 4. **Calculate squares:** $$27225 + x^2 = 29929$$ 5. **Isolate $x^2$:** $$x^2 = 29929 - 27225$$ $$x^2 = 2704$$ 6. **Find $x$ by taking the square root:** $$x = \sqrt{2704}$$ $$x = 52$$ 7. **Answer:** The length of the side between Dayton and Lima is $52$ kilometers.