The distance from a point to a line

The distance from a point to a line is the shortest gap, measured straight across (perpendicular) from the point to the line. Here we find the distance from the origin $O(0, 0)$ to the line $3x + 4y - 25 = 0$.

In general, the distance from a point $(x_0, y_0)$ to a line $ax + by + c = 0$ is given by this formula.

For this line $a = 3$, $b = 4$, $c = -25$. Substituting $(0, 0)$, the numerator is $|3 \cdot 0 + 4 \cdot 0 - 25| = 25$ and the denominator is $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$, so the distance is $d = \dfrac{25}{5} = 5$.

The perpendicular dropped from $O$ to the line is $y = \dfrac{4}{3}x$, and it meets the line at the foot of the perpendicular $(3, 4)$. The length from the origin to $(3, 4)$ is $\sqrt{3^2 + 4^2} = 5$, matching the distance from the formula.

This formula is exactly what lies behind the earlier topic "intersection of a circle and a line," where we compared the distance from the center to the line with the radius. The circle $x^2 + y^2 = 25$ (center $(0, 0)$, radius $5$) is tangent to this line, since the distance is $5$, equal to the radius. The large dots on the graph are the point $O(0, 0)$ and the foot of the perpendicular $(3, 4)$; the length of the segment joining them is the distance we wanted.