Parallel and perpendicular lines
Practice Flashcards
Flip to reveal answersWhat is the condition for two lines to be parallel?
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All 16 Flashcards — Parallel and perpendicular lines
Sign up free to track progress and get spaced-repetition review schedules.
Question
What is the condition for two lines to be parallel?
Answer
Two lines are parallel if and only if they have the same gradient. They never intersect (unless they are the same line). Example: y = 3x + 2 and y = 3x − 7 are parallel — both have m = 3.
Question
Line L₁ has gradient m. State the gradient of any line parallel to L₁.
Answer
Any line parallel to L₁ also has gradient m. The gradient is the same — only the y-intercept (c) can differ.
Question
Are y = −2x + 5 and y = −2x − 3 parallel? Explain why.
Answer
Yes — both have gradient m = −2. They are different lines (different y-intercepts: 5 and −3), so they are parallel, not the same line.
Question
Exam trap: A student sees y = 2x + 1 and y = −2x + 1 and says they are parallel because "they look similar." Are they parallel?
Answer
No — gradients are +2 and −2. These are different gradients, so the lines are not parallel. They intersect at (0, 1). Similar equations do not mean parallel lines — the gradient values must match exactly.
Question
What is the condition for two lines to be perpendicular?
Answer
Two lines are perpendicular if the product of their gradients equals −1: m₁ × m₂ = −1. This means the gradients are negative reciprocals of each other.
Question
If a line has gradient m, state the gradient of a line perpendicular to it.
Answer
The perpendicular gradient is −1/m (flip the fraction and change the sign). Examples: m = 3 → m⊥ = −1/3 m = −2/5 → m⊥ = 5/2 m = 4 → m⊥ = −1/4
Question
A line has gradient −3/4. Find the gradient of a perpendicular line.
Answer
m⊥ = −1 / (−3/4) = 4/3. Rule: flip the fraction (4/3) and change the sign. Starting negative → perpendicular is positive. Check: (−3/4) × (4/3) = −12/12 = −1 ✓
Question
Exam trap: A line has gradient 5. A student says the perpendicular gradient is −5. What is the error?
Answer
They only changed the sign but did not take the reciprocal. The perpendicular gradient is −1/5 (flip to 1/5, then negate). "Negative reciprocal" means both steps: flip AND change sign.
Question
Describe the method to find the equation of a line parallel to y = 4x − 1 through the point (3, 7).
Answer
1. Identify the gradient: m = 4 (same as the original line — parallel). 2. Substitute into y = 4x + c using (3, 7): 7 = 4(3) + c → c = −5. 3. Equation: y = 4x − 5.
Question
Find the equation of the line perpendicular to y = 2x + 3 that passes through (4, 1).
Answer
m⊥ = −1/2. y = −(1/2)x + c. Use (4, 1): 1 = −(1/2)(4) + c → 1 = −2 + c → c = 3. Equation: y = −(1/2)x + 3.
Question
A line L₁ has equation y = −3x + 2. Find the equation of the line L₂, perpendicular to L₁, that passes through (0, 5).
Answer
m⊥ = 1/3 (negative reciprocal of −3). The line passes through (0, 5), so c = 5 directly (it is the y-intercept). Equation of L₂: y = (1/3)x + 5.
Question
Exam trap: When writing a perpendicular line equation, a student uses the original gradient from the question instead of the negative reciprocal. What is the consequence?
Answer
Their answer will be a parallel line, not a perpendicular one — a completely different type of answer. Always find m⊥ = −1/m first, before substituting the given point to find c.
Question
What is the perpendicular bisector of a line segment AB?
Answer
The perpendicular bisector is a line that: 1. Passes through the midpoint of AB. 2. Is perpendicular to AB (i.e. meets AB at a right angle). Every point on the perpendicular bisector is equidistant from A and B.
Question
What two things do you need in order to write the equation of the perpendicular bisector of segment AB?
Answer
1. The midpoint of AB — the perpendicular bisector passes through this point. 2. The perpendicular gradient — find the gradient of AB first, then take the negative reciprocal.
Question
Find the equation of the perpendicular bisector of the segment joining A(2, 4) and B(6, 8).
Answer
Midpoint M = ((2+6)/2, (4+8)/2) = (4, 6). Gradient of AB: m = (8−4)/(6−2) = 4/4 = 1. So m⊥ = −1. y = −x + c. Use (4, 6): 6 = −4 + c → c = 10. Perpendicular bisector: y = −x + 10.
Question
Exam trap: When finding a perpendicular bisector, a student finds the midpoint correctly but then uses one of the original endpoints to find c instead of the midpoint. What goes wrong?
Answer
The line will pass through the wrong point — it will be perpendicular to AB but not at the midpoint. The perpendicular bisector must pass through the midpoint, not through A or B. Always substitute the midpoint to find c.
Read the notes
Full study notes for Parallel and perpendicular lines
Topic 2.1 hub
Equations of a line
More from Topic 2.1
All flashcards in this topic
Math AI SL exam skills
Paper structures & tips
Track your progress with spaced repetition
Sign up free — Aimnova tells you exactly which cards to review and when, so you remember everything before your IB exam.
Start Free