Question: Pretest: Unit 3
A walking path across a park is represented by the equation $y = -2x + 5$. A new path will be built perpendicular to this path. The paths will intersect at the point $(-2, 9)$. Identify the equation that represents the new path.
Graph: a coordinate grid with a downward-sloping line passing through the center-right area; the line is steep and crosses near the y-axis above 0 and the x-axis just to the right of the origin. position_hint = center
A. $y = \frac{1}{2} x + 10$
1. **State the problem:** We have a path represented by the equation $$y = -2x + 5$$ and a new path perpendicular to it passing through the point $$(-2, 9)$$. We need to find the equation of the new path.
2. **Recall the rule for perpendicular lines:** The slope of a line perpendicular to another line with slope $$m$$ is $$-\frac{1}{m}$$.
3. **Identify the slope of the original path:** The slope $$m_1$$ of the original path is $$-2$$.
4. **Calculate the slope of the new path:** Using the perpendicular slope formula,
$$m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2}$$.
5. **Use point-slope form to find the new path's equation:** The point-slope form is
$$y - y_1 = m(x - x_1)$$,
where $$m = \frac{1}{2}$$ and the point is $$(-2, 9)$$.
6. **Substitute values:**
$$y - 9 = \frac{1}{2}(x - (-2))$$
$$y - 9 = \frac{1}{2}(x + 2)$$
7. **Simplify:**
$$y - 9 = \frac{1}{2}x + 1$$
8. **Add 9 to both sides:**
$$y = \frac{1}{2}x + 1 + 9$$
$$y = \frac{1}{2}x + 10$$
9. **Final answer:** The equation of the new path is $$y = \frac{1}{2}x + 10$$.
This matches option A.