Perpendicular to a Line
Grade 7 Geometry — A Complete Guide
1. Introduction
Imagine you're standing on a road and want to find the shortest path to a river. Would you walk diagonally, or would you walk straight toward the river at a right angle? The answer involves one of geometry's most important concepts: the perpendicular from a point to a line.
This concept:
- Connects a point with a line in a very specific way
- Creates a special segment that forms a right angle (90°) with the line
- Is the fundamental tool for measuring distance from a point to a line
- Builds on your knowledge of perpendicular lines
Interactive visualization — available in the full lesson
2. Definition of Perpendicular from a Point
Definition: A perpendicular from a point to a line is a segment drawn from the point to the line such that it forms a right angle (90°) with the line.
Key features:
- The segment is perpendicular to the given line
- It creates a right angle at the point of intersection
- We write $AH \perp l$ to mean "segment $AH$ is perpendicular to line $l$"
3. Foot of the Perpendicular
Definition: The point where the perpendicular meets the line is called the foot of the perpendicular.
If the perpendicular from point $A$ meets line $l$ at point $H$, then:
- $H$ is the foot of the perpendicular
- $H$ is also called the base point
- $H$ lies on line $l$
- $\angle AHl = 90°$
4. Formal Definition
📐 Formal Definition
Let point $A$ be a point outside line $l$. The segment $\overline{AH}$ is called the perpendicular from $A$ to $l$ if:
- $H$ is a point on line $l$
- $\overline{AH}$ forms a right angle with $l$, i.e., $\angle AHB = 90°$ for any other point $B$ on $l$
We write: $\boxed{AH \perp l}$, and $H$ is the foot of the perpendicular.