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
- 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.
- 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 $2x – 3y + 5 = 0$
Identify the Components
In the line equation $2x – 3y + 5 = 0$:- $A = 2$
- $B = -3$
- $C = 5$
The point is $(x_1, y_1) = (3, 4)$
- Substitute into the Formula
Plug these values into the distance formula:$d = frac{|2(3) – 3(4) + 5|}{sqrt{2^2 + (-3)^2}}$
- Simplify the Numerator
Calculate the numerator:$|2(3) – 3(4) + 5| = |6 – 12 + 5| = |-1| = 1$
- Simplify the Denominator
Calculate the denominator:$sqrt{2^2 + (-3)^2} = sqrt{4 + 9} = sqrt{13}$
- 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.