How to Find the Distance to a Line?

Finding the distance from a point to a line is a common problem in geometry. This distance is the shortest length from the point to any point on the line, measured perpendicularly.

The Formula

If you have a line given by the equation $Ax + By + C = 0$ and a point $(x_1, y_1)$, the distance from the point to the line can be found using the following formula:

$d = frac{|Ax_1 + By_1 + C|}{sqrt{A^2 + B^2}}$

Breaking Down the Formula

1. Numerator: $|Ax_1 + By_1 + C|$ – This part calculates the absolute value of the line equation when the point $(x_1, y_1)$ is substituted into it.
2. Denominator: $sqrt{A^2 + B^2}$ – This part normalizes the equation, ensuring that the distance is correctly scaled.

Step-by-Step Example

Let’s go through a practical example to make this clearer.

Example Problem

Find the distance from the point $(3, 4)$ to the line $2x – 3y + 5 = 0$

1. Identify the Components
   In the line equation $2x – 3y + 5 = 0$:

   • $A = 2$
   • $B = -3$
   • $C = 5$

   The point is $(x_1, y_1) = (3, 4)$

2. Substitute into the Formula
   Plug these values into the distance formula:

   $d = frac{|2(3) – 3(4) + 5|}{sqrt{2^2 + (-3)^2}}$

3. Simplify the Numerator
   Calculate the numerator:

   $|2(3) – 3(4) + 5| = |6 – 12 + 5| = |-1| = 1$

4. Simplify the Denominator
   Calculate the denominator:

   $sqrt{2^2 + (-3)^2} = sqrt{4 + 9} = sqrt{13}$

5. Calculate the Distance
   Combine the results to find the distance:

   $d = frac{1}{sqrt{13}} approx 0.28$

Conclusion

The distance from the point $(3, 4)$ to the line $2x – 3y + 5 = 0$ is approximately $0.28$ units. Understanding this formula and the steps involved can help you solve similar problems in geometry and real-world applications, such as finding the shortest path in navigation or computer graphics.

Citations

1. 1. Khan Academy – Distance from a point to a line
2. 2. Math is Fun – Distance from a Point to a Line
3. 3. Paul’s Online Math Notes