MathematicsChapter 8: Geometry of Straight Lines
Parallel Perpendicular Collinear
Slope compares lines without drawing them to scale.
Learning objectives
- Explain parallel perpendicular collinear in clear mathematical language
- Apply the ideas in parallel and perpendicular
- Solve problems involving collinear points
Definition
Slope compares lines without drawing them to scale. Two non-vertical lines are parallel when m1 = m2 (equal gradients, so equal inclination). They are perpendicular when m1 m2 = -1, that is, each slope is the negative reciprocal of the other. Three points are collinear when the slopes of AB, BC, and AC are equal (or all undefined on one vertical line). Equal slopes mean the segments continue the same tilt.
Mathematical form
m1 = m2 (equal gradients
m2 = -1
m=0 is perpendicular to a vertical line
Where / symbols
- parallel
- Two non-vertical lines are parallel when m1 = m2 (equal gradients, so equal inclination).
- m1 = m2
- Two non-vertical lines are parallel when m1 = m2 (equal gradients, so equal inclination).
- perpendicular
- They are perpendicular when m1 m2 = -1, that is, each slope is the negative reciprocal of the other.
- m1 m2 = -1
- They are perpendicular when m1 m2 = -1, that is, each slope is the negative reciprocal of the other.
Relationship
- m1 = m2
- m1 m2 = -1
Overview
Same tilt, right angle, or one line:
Parallel and Perpendicular
Parallel lines never meet: for finite slopes, m1 = m2. Two vertical lines are also parallel (both have undefined slope). Perpendicular lines meet at 90 deg: m1 m2 = -1, so m2 = -1m1, the negative reciprocal. If one line is vertical, the perpendicular is horizontal (m=0), and the product rule is not used because undefined times 0 is not a real product.
Examples m=2 and m=2 parallel · m=2 and m=-1/2 perpendicular · Vertical with horizontal: 90 deg
Collinear Points
Points A, B, C are collinear when AB, BC, and AC have equal slopes (including the all-vertical case). You only need two of those slopes to match, then the third follows. If slope AB is 1 and slope BC is 2, C is off the line of AB. This test agrees with last chapter's distance test AC=AB+BC when B is between A and C, but slope does not care who sits in the middle.
Examples (0,0), (1,2), (2,4) slopes all 2 · (1,1), (1,3), (1,8) all vertical · Unequal slopes: not collinear
What it means
Slope compares lines without drawing them to scale.
Main ideas
1. PARALLEL AND PERPENDICULAR: Parallel lines never meet: for finite slopes, m1 = m2. 2. COLLINEAR POINTS: Points A, B, C are collinear when AB, BC, and AC have equal slopes (including the all-vertical case).
Why it matters
Negative reciprocal means flip the fraction and change the sign: 34 becomes -43. Parallel is equal m, not negative equal. A line with m=0 is perpendicular to a vertical line, not to another horizontal line.
How to answer in an exam
Parallel: m1=m2. Perpendicular: m1 m2 = -1 (negative reciprocal), except the vertical-horizontal pair. Collinear points: equal slopes of AB, BC, and AC. Do not use m1 m2 = 1; that is not a right angle.
Explanation
Negative reciprocal means flip the fraction and change the sign: 34 becomes -43. Parallel is equal m, not negative equal. A line with m=0 is perpendicular to a vertical line, not to another horizontal line.
In simple words
Parallel: m1=m2. Perpendicular: m1 m2 = -1 (negative reciprocal), except the vertical-horizontal pair. Collinear points: equal slopes of AB, BC, and AC. Do not use m1 m2 = 1; that is not a right angle.
Key points
- parallel
- m1 = m2
- perpendicular
- m1 m2 = -1
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Parallel Perpendicular Collinear
Sticky Notes · Quick Revision
Core Idea
- Slope compares lines without drawing them to scale.
Key Terms
- parallel: Two non-vertical lines are parallel when m1 = m2 (equal gradients, so equal inclination).
- m1 = m2: Two non-vertical lines are parallel when m1 = m2 (equal gradients, so equal inclination).
Main Pattern
- 1. PARALLEL AND PERPENDICULAR: Parallel lines never meet: for finite slopes, m1 = m2.
- 2. COLLINEAR POINTS: Points A, B, C are collinear when AB, BC, and AC have equal slopes (including the all-vertical case).
Worked Reminder
- m=2 and m=2 parallel
- m=2 and m=-12 perpendicular
- Vertical with horizontal: 90 deg
Common Mistake
- Parallel: m1=m2. Perpendicular: m1 m2 = -1 (negative reciprocal), except the vertical-horizontal pair. Collinear points: equal slopes of AB, BC, and AC. Do not use m1 m2 = 1; that is not a right angle.
Exam Tip
- State the rule for parallel perpendicular collinear, show each algebraic step, and check the final answer against the question.
60-second revision
- Slope compares lines without drawing them to scale.
- Parallel lines never meet: for finite slopes, m1 = m2.
- Points A, B, C are collinear when AB, BC, and AC have equal slopes (including the all-vertical case).
- Negative reciprocal means flip the fraction and change the sign: 34 becomes -43.
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