Question: The point A has coordinates $(0, 0)$ and the point B has coordinates $(8, -16)$. The line $l_1$ passes through $B$ and is perpendicular to $AB$. Find an equation for the line $l_1$. Give your answer in the form $y = mx + c$

1. **State the problem:** We need to find the equation of the line $l_1$ that passes through point $B(8, -16)$ and is perpendicular to the line segment $AB$ where $A = (0,0)$ and $B = (8,-16)$. 2. **Find the slope of line $AB$:** The slope formula is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using points $A(0,0)$ and $B(8,-16)$: $$m_{AB} = \frac{-16 - 0}{8 - 0} = \frac{-16}{8} = -2$$ 3. **Find the slope of line $l_1$:** Since $l_1$ is perpendicular to $AB$, its slope $m_{l_1}$ is the negative reciprocal of $m_{AB}$: $$m_{l_1} = -\frac{1}{m_{AB}} = -\frac{1}{-2} = \frac{1}{2}$$ 4. **Use point-slope form to find the equation of $l_1$:** The point-slope form is: $$y - y_1 = m(x - x_1)$$ Using point $B(8, -16)$ and slope $m_{l_1} = \frac{1}{2}$: $$y - (-16) = \frac{1}{2}(x - 8)$$ Simplify: $$y + 16 = \frac{1}{2}x - 4$$ 5. **Solve for $y$ to get slope-intercept form $y = mx + c$:** $$y = \frac{1}{2}x - 4 - 16$$ $$y = \frac{1}{2}x - 20$$ **Final answer:** $$\boxed{y = \frac{1}{2}x - 20}$$