Distance from a Point to a Line.
Distance Between Parallel Lines
Geometry โ Grade 7
๐ 1. Introduction: From Distance Between Points to Distance from a Point to a Line
You already know the most basic distance concept in geometry:
Definition: The distance between two points is the length of the segment connecting these two points.
For example, the distance between points $A$ and $B$ is the length of segment $AB$.
But what if we want to measure the distance from a point to an entire line? A line has infinitely many points โ which one do we measure to? This is the question we answer in this lesson.
The key idea is the perpendicular. Let's explore why.
๐ 2. Perpendicular and Oblique Line
Perpendicular from a Point to a Line
Definition: Let point $A$ be a point not on line $a$. The segment $AH$, where $H$ is the point on line $a$ such that $AH \perp a$, is called the perpendicular from point $A$ to line $a$. Point $H$ is called the foot of the perpendicular.
Oblique Line
Definition: The segment $AM$, where $M$ is any point on line $a$ different from $H$ (i.e., $M \neq H$), is called an oblique line (or simply an oblique) from point $A$ to line $a$.
Projection of the Oblique Line
Definition: The segment $HM$ on line $a$ โ from the foot of the perpendicular $H$ to the foot of the oblique line $M$ โ is called the projection of the oblique line $AM$ onto line $a$.
Interactive visualization โ available in the full lesson
๐บ The Right Triangle Relationship
Look at triangle $AHM$. Since $AH \perp a$, the angle at $H$ is $90ยฐ$. So triangle $AHM$ is a right triangle where:
- $AH$ is a leg (cathetus)
- $HM$ is the other leg
- $AM$ is the hypotenuse (the side opposite the right angle)
Key Property: In any right triangle, the hypotenuse is longer than either leg. Therefore:
$$AH < AM$$
The perpendicular $AH$ is shorter than the oblique line $AM$.