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Parallel Perpendicular Collinear
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 s
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Parallel Perpendicular Collinear
What is it?
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.
In simple words
- 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.
Important 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).
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.
What you should learn
- Explain parallel perpendicular collinear in clear mathematical language
- Apply the ideas in parallel and perpendicular
- Solve problems involving collinear points
Examples
- m=2 and m=2 parallel
- m=2 and m=-12 perpendicular
- Vertical with horizontal: 90 deg
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.
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.
Recommended for Class 9
Full school-level explanation
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.
Same tilt, right angle, or one line:
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.
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.
Remember
- m1 = m2
- m1 m2 = -1
- negative reciprocal
Examples
- m=2 and m=2 parallel
- m=2 and m=-12 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.
Remember
- collinear
- equal slopes
- AB, BC, and AC
Examples
- (0,0), (1,2), (2,4) slopes all 2
- (1,1), (1,3), (1,8) all vertical
- Unequal slopes: not collinear
Remember
- parallel
- m1 = m2
- perpendicular
- m1 m2 = -1
- negative reciprocal
- collinear
- equal slopes
- flip the fraction
How the ideas connect
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.
Deeper understanding
Go beyond memorising Parallel Perpendicular Collinear. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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.
Common mistakes
Exam focus
Core Idea
Key Terms
Main Pattern
Worked Reminder
Exam focus
Quick revision
- parallel
- m1 = m2
- perpendicular
- m1 m2 = -1
- 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).
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.
For long answers on Parallel Perpendicular Collinear: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- parallel
- m1 = m2
- perpendicular
- m1 m2 = -1
Important formulas
- parallel
- m1 = m2
- perpendicular
Common exam traps
- Parallel: m1=m2.
- Perpendicular: m1 m2 = -1 (negative reciprocal), except the vertical-horizontal pair.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Parallel: m1=m2.
- Mistake
Perpendicular: m1 m2 = -1 (negative reciprocal), except the vertical-horizontal pair.
- Mistake
Do not mix up parallel and perpendicular with collinear points; use the condition given in the question.
- 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
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
parallel
m1 = m2
perpendicular
m1 m2 = -1
Slope compares lines without drawing them to scale. Two non-vertical lines are parallel when m1 = m2 (equal gradients, so equal inclination). They are perpendic
m1 = m2
- 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 perpendic
- Formula
- m1 = m2
- Example
- m=2 and m=2 parallel
- Critical warning
- Parallel: m1=m2.
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Exam Practice
Define
Q1. Define Parallel Perpendicular Collinear.
Explain
Q2. Explain: PARALLEL AND PERPENDICULAR.
Example
Q3. Give one clear example of Parallel Perpendicular Collinear and say why it fits.
Mistake
Q4. State one common mistake students make about Parallel Perpendicular Collinear.
Exam note
Q5. Write a short exam note on Parallel Perpendicular Collinear: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.