Find the distance between a point and a line

Learning Objectives Define the distance between a point and a line as the length of the perpendicular segment from the point to the line. Find the equation of a line perpendicular to a given line that passes through a specific point. Calculate the point of intersection between two perpendicular lines by solving a system of equations. Apply the distance formula to find the length of the segment connecting a point to its perpendicular intersection on a line. Use the specialized formula to efficiently calculate the distance between a point and a line in standard form (Ax + By + C = 0). Solve geometric problems involving the shortest distance from a point to a line. How does a self-driving car know the exact distance to the edge of its lane? 🚗 It's all about finding the sh...

TermDefinitionExample Perpendicular DistanceThe shortest distance from a point to a line. It is measured along the line segment that is perpendicular to the original line and connects it to the point.Imagine standing in a field and wanting to walk to a straight road. The shortest path isn't diagonal; it's the one that makes a 90-degree angle with the road. Perpendicular LinesTwo lines that intersect to form a right (90°) angle. Their slopes are negative reciprocals of each other.If a line has a slope of 2, any line perpendicular to it will have a slope of -1/2. Negative ReciprocalA number that, when multiplied by the original number, equals -1. To find it, you flip the fraction and change the sign.The negative reciprocal of 3/4 is -4/3. The negative reciprocal of -5 is 1/5. Poin...

Perpendicular Slope Rule If a line has slope `m`, a line perpendicular to it has a slope of `m_{\perp} = -\frac{1}{m}`. Use this rule to find the slope of the perpendicular line needed to calculate the shortest distance. This is the first step in the geometric method. Distance Formula Between Two Points d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Use this formula to find the distance between the original point and the point of intersection you calculate on the line. Distance from a Point to a Line Formula The distance `d` from a point `(x_1, y_1)` to the line `Ax + By + C = 0` is: `d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}` This is a direct formula for finding the distance. It is very efficient but requires the line's equation to be in standard form.