Perpendicular Distance from Point to Line Formula

Answer

The image displays the Perpendicular Distance Formula, which calculates the shortest distance $d$ from a specific point $P(x_1, y_1)$ to a line $L$ defined by the general equation $Ax + By + C = 0$. This distance is measured along a line segment through $P$ that is perpendicular to $L$.

Observation

The provided diagram illustrates a Cartesian coordinate system with a line $L$ having a negative slope. A point $P$ is located in the first quadrant, and a perpendicular line segment of length $d$ connects $P$ to the line $L$. The formula for this distance is explicitly stated in the top right corner.

Explanation

1. Identify the Components To use this formula, you must have the line in its "General Form" ($Ax + By + C = 0$) and the coordinates of the external point $P(x_1, y_1)$. ⚠️ This step is required on exams: always rearrange $y = mx + b$ into $Ax + By + C = 0$ before starting.

2. Substitute the Point into the Line Equation We "test" how far the point is from the line by substituting $x_1$ and $y_1$ into the linear expression. Since distance cannot be negative, we apply absolute value bars. $|Ax_1 + By_1 + C|$ This expression represents the magnitude of the "offset" of the point from the line.

3. Normalize by the Magnitude of the Normal Vector The coefficients $A$ and $B$ represent the components of the normal vector (a vector perpendicular to the line). To get the actual geometric distance, we divide by the length of this vector. $\sqrt{A^2 + B^2}$ This is the Pythagorean length of the vector perpendicular to the line.

4. Combine into the Final Formula Putting it all together, we arrive at the standard formula shown in your image. $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ The final formula calculates the shortest (perpendicular) distance between a point and a line in 2D space.

Final Answer

The formula for the perpendicular distance $d$ from point $P(x_1, y_1)$ to line $Ax + By + C = 0$ is: $\boxed{d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}}$

Common Mistakes

• Incorrect Signage: Forgetting to move all terms to one side of the equation (making sure the equation equals zero) before identifying $A$, $B$, and $C$. If the line is $y = 2x + 3$, you must rewrite it as $-2x + y - 3 = 0$.
• Denominator Error: Some students accidentally include $C$ under the square root in the denominator; remember that only the coefficients of $x$ and $y$ ($A$ and $B$) are used to find the magnitude of the normal vector.