Coordinate geometry · Ages 14–15
Slopes of parallel and perpendicular lines
Parallel: equal slopes; perpendicular: product −1.
How to do it
Method
Parallel: equal slopes; perpendicular: product −1
23 × (−32) = −1
Another way
Turn it over, change the sign
23 → 32 → −32
In full
Parallel lines have equal slopes: m₁ = m₂. Perpendicular lines have slopes multiplying to −1: m₁ × m₂ = −1 (neither line horizontal nor vertical). A line of slope 23 has perpendiculars of slope m with 23 × m = −1, so m = −32. Another way: turn it over and change the sign: 23 upside down is 32, with the sign changed −32, the same answer. To decide how two lines meet, find each slope: equal, parallel; multiplying to −1, perpendicular; neither, they simply cross. Careful: slopes 2 and −2 are neither, since 2 × (−2) = −4, not −1. A horizontal line (slope 0) and a vertical one (no slope at all) are perpendicular too, but the product test can't show it.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Line L₁ has slope 54, and line L₂ is perpendicular to L₁. L₂'s slope =
Answer in simplest form
Show the answer and working
Answer: −45
54 × m₂ = −1, so m₂ = −45: turn 54 over and change its sign.
Example 2
Line L₁ passes through A(−6, −1) and B(−1, −3), and line L₂ is parallel to L₁. L₂'s slope =
Answer in simplest form
Show the answer and working
Answer: −25
Parallel lines have equal slopes: L₁'s is −25 = −25, so L₂'s is −25.
Example 3
What is the slope of a line perpendicular to one of slope −43?
- A. −34
- B. 34
- C. −43
- D. 43
Choose the right answer
Show the answer and working
Answer: B. 34
Perpendicular slopes multiply to −1: −43 × m = −1, so m is −43 turned over with its sign changed.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Line L₁ passes through A(−4, 2) and B(0, −6), and line L₂ is parallel to L₁. L₂'s slope =
Answer in simplest form
Answer:
2.What is the slope of a line perpendicular to one of slope 51?
- A. 15
- B. −51
- C. −15
- D. 51
Choose the right answer
Answer:
3.Lines of slopes −52 and 25 are perpendicular.
True or false?
Answer:
4.A(−4, −3), B(−7, −6), C(2, 4), D(−4, k), and AB ∥ CD. k =
Answer:
5.L₁ passes through A(−5, −2) and B(−4, −4), L₂ through C(−5, 5) and D(−9, 7). L₁'s slope = , L₂'s slope = , and their product = .
Fill in each blank
Answer:
All the answers
- 1. −2
- 2. C. −15
- 3. True
- 4. −2
- 5. −2, −12, 1
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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