Horizontal Vertical Collinear

Horizontal Vertical Collinear

What is it?

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.

In simple words

• 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.

Important 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|.

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|.

What you should learn

• Explain horizontal vertical collinear in clear mathematical language
• Apply the ideas in horizontal and vertical
• Solve problems involving collinear or a triangle

Examples

• (2,5) to (7,5) is 5
• (3,1) to (3,-4) is 5
• (0,2) to (0,2) is 0

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.

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.

Full school-level explanation

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.

Special segments first, then three points on a line:

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.

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.

Remember

• y1=y2
• |x2-x1|
• x1=x2
• |y2-y1|

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.

Remember

• Collinear
• AC = AB+BC
• non-collinear

Examples

• (0,0), (1,1), (2,2) collinear
• AC=AB+BC when B is between
• (0,0), (1,0), (0,1) a triangle

Remember

• horizontal
• |x2-x1|
• vertical
• |y2-y1|
• collinear
• AC = AB+BC
• non-collinear
• collinear by slopes

How the ideas connect

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.

Deeper understanding

Go beyond memorising Horizontal Vertical Collinear. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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.

Common mistakes

Exam focus

Core Idea

Key Terms

Main Pattern

Worked Reminder

Exam focus

Quick revision

• horizontal
• |x2-x1|
• vertical
• |y2-y1|
• 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|.

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.

For long answers on Horizontal Vertical Collinear: define clearly → explain the main parts → give one example → state one exam caution.