Perpendicular Line Cc2194

1. **State the problem:** We need to find the equation of a line perpendicular to $y = -2x + 4$ that passes through the point $(4, 2)$. 2. **Recall the slope of the given line:** The given line is in slope-intercept form $y = mx + b$ where $m$ is the slope. Here, $m = -2$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, the perpendicular slope $m_{\perp} = -\frac{1}{m} = -\frac{1}{-2} = \frac{1}{2}$. 4. **Use point-slope form to find the equation:** The point-slope form is $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1) = (4, 2)$ and $m = \frac{1}{2}$. Substitute: $$y - 2 = \frac{1}{2}(x - 4)$$ 5. **Simplify the equation:** $$y - 2 = \frac{1}{2}x - 2$$ Add 2 to both sides: $$y = \frac{1}{2}x - 2 + 2$$ $$y = \frac{1}{2}x$$ 6. **Check the options:** - A: $y = -\frac{1}{2} x + 2$ (slope $-\frac{1}{2}$, not $\frac{1}{2}$) - B: $y = \frac{1}{2} x$ (matches our equation) - C: $y = \frac{1}{2} x + 4$ (correct slope but wrong intercept) - D: $y = -2x$ (same slope as original, not perpendicular) **Final answer:** Option B, $y = \frac{1}{2} x$.