MathematicsChapter 7: Coordinate Geometry
Horizontal Vertical Collinear
Two special distances skip the full formula.
Learning objectives
- Explain horizontal vertical collinear in clear mathematical language
- Apply the ideas in horizontal and vertical
- Solve problems involving collinear or a triangle
Definition
Two special distances skip the full formula. If the segment is horizontal, y-coordinates match and distance is |x2-x1|. If it is vertical, x-coordinates match and distance is |y2-y1|. Three points are collinear when they lie on one straight line. If B lies between A and C, then AC = AB+BC. The vertices of a triangle must be non-collinear: if they line up, the figure has no area and is not a triangle.
Formula
AC = AB+BC. The vertices of a triangle must b
Where / symbols
- horizontal
- If the segment is horizontal, y-coordinates match and distance is |x2-x1|.
- |x2-x1|
- If the segment is horizontal, y-coordinates match and distance is |x2-x1|.
- vertical
- If it is vertical, x-coordinates match and distance is |y2-y1|.
- |y2-y1|
- If it is vertical, x-coordinates match and distance is |y2-y1|.
Relationship
- AC = AB+BC
Overview
Special segments first, then three points on a line:
Horizontal and Vertical
Horizontal means the same height: y1=y2, so the rise is 0 and distance = |x2-x1|. Vertical means the same run: x1=x2, so distance = |y2-y1|. Example: (2,5) to (7,5) is 5. Example: (3,1) to (3,-4) is 5. Using the full sqrt formula still works, but you will take sqrt of a square and get the same absolute value.
Examples (2,5) to (7,5) is 5 · (3,1) to (3,-4) is 5 · (0,2) to (0,2) is 0
Collinear Or A Triangle
Collinear points sit on one line. A practical test: B is between A and C when AC = AB+BC (the long side equals the sum of the two short ones). If the three distances fail that additive check in every order, the points are not collinear. Triangle vertices must be non-collinear; otherwise AB+BC=AC and the path is a segment, not a closed triangle with positive area.
Examples (0,0), (1,1), (2,2) collinear · AC=AB+BC when B is between · (0,0), (1,0), (0,1) a triangle
What it means
Two special distances skip the full formula.
Main ideas
1. HORIZONTAL AND VERTICAL: Horizontal means the same height: y1=y2, so the rise is 0 and distance = |x2-x1|. 2. COLLINEAR OR A TRIANGLE: Collinear points sit on one line.
Why it matters
You can also test collinear by slopes (next chapter): the slope of AB equals the slope of BC. Distance addition is enough when you already computed the three lengths. For a triangle, confirm non-collinear before you quote a perimeter or an area.
How to answer in an exam
Horizontal is |x2-x1|, not the full hypotenuse guess. Vertical is |y2-y1|. Collinear with B between A and C means AC=AB+BC. A triangle needs non-collinear vertices; three points on a line do not make triangle ABC.
Explanation
You can also test collinear by slopes (next chapter): the slope of AB equals the slope of BC. Distance addition is enough when you already computed the three lengths. For a triangle, confirm non-collinear before you quote a perimeter or an area.
In simple words
Horizontal is |x2-x1|, not the full hypotenuse guess. Vertical is |y2-y1|. Collinear with B between A and C means AC=AB+BC. A triangle needs non-collinear vertices; three points on a line do not make triangle ABC.
Key points
- horizontal
- |x2-x1|
- vertical
- |y2-y1|
- y1=y2
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Horizontal Vertical Collinear
Sticky Notes · Quick Revision
Core Idea
- Two special distances skip the full formula.
Key Terms
- horizontal: If the segment is horizontal, y-coordinates match and distance is |x2-x1|.
- |x2-x1|: If the segment is horizontal, y-coordinates match and distance is |x2-x1|.
Main Pattern
- 1. HORIZONTAL AND VERTICAL: Horizontal means the same height: y1=y2, so the rise is 0 and distance = |x2-x1|.
- 2. COLLINEAR OR A TRIANGLE: Collinear points sit on one line.
Worked Reminder
- (2,5) to (7,5) is 5
- (3,1) to (3,-4) is 5
- (0,2) to (0,2) is 0
Common Mistake
- Horizontal is |x2-x1|, not the full hypotenuse guess. Vertical is |y2-y1|. Collinear with B between A and C means AC=AB+BC. A triangle needs non-collinear vertices; three points on a line do not make triangle ABC.
Exam Tip
- State the rule for horizontal vertical collinear, show each algebraic step, and check the final answer against the question.
60-second revision
- Two special distances skip the full formula.
- Horizontal means the same height: y1=y2, so the rise is 0 and distance = |x2-x1|.
- Collinear points sit on one line.
- You can also test collinear by slopes (next chapter): the slope of AB equals the slope of BC.
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