# [[Perpendicular Distance from a Point to a Line]] (Mathematics > Geometry) ## Definition The shortest distance between a point and a line occurs along the line that is perpendicular to the given line and passes through the point. The point of intersection of this perpendicular line with the original line is the closest point. This is used to calculate the shortest distance between the point and the line. ## Key Concepts - The shortest distance between a point and a line occurs along a perpendicular line to the given line. - The slope of the perpendicular line is the negative reciprocal of the slope of the original line. - The point on the line that is closest to the given point is found by solving for the intersection of the perpendicular and original lines. ## Important Properties 1. The slope of the perpendicular line to $y = mx + b$ is $-\frac{1}{m}$. 2. The intersection point of the original line and the perpendicular line gives the closest point. 3. The distance is the length of the perpendicular segment between the point and the line. ## Essential Formulas - For a line with equation $y = mx + b$, the perpendicular line passing through the point $(x_1, y_1)$ has the equation: $y - y_1 = -\frac{1}{m}(x - x_1)$ - Distance between a point $(x_1, y_1)$ and the line $Ax + By + C = 0$: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ ## Core Examples 1. **Finding the closest point**: Given the line $y = 2x + 1$ and the point $(3, 5)$, the slope of the perpendicular line is $-\frac{1}{2}$. The equation of the perpendicular line is: $y - 5 = -\frac{1}{2}(x - 3)$ Solving the system of equations gives the closest point. 2. **Distance from a point to a line**: For the point $(1, 2)$ and the line $x + 2y - 3 = 0$, use the distance formula to find the shortest distance: $d = \frac{|1(1) + 2(2) - 3|}{\sqrt{1^2 + 2^2}} = \frac{2}{\sqrt{5}} \approx 0.89$ ## Related Theorems/Rules - [[Perpendicular Line Theorem]] - [[Slope of Perpendicular Lines]] ## Common Pitfalls - Incorrectly calculating the slope of the perpendicular line. - Forgetting to solve for the intersection point when finding the closest point on the line. ## Related Topics - [[Point-Slope Form of a Line]] - [[Linear Equations and Slopes]] ## Quick Review Questions 1. What is the equation of the line perpendicular to $y = 3x + 2$ that passes through the point $(4, -1)$? 2. How do you find the point on a line that is closest to a given point?