Coordinate geometry · Ages 14–15
Using slopes
In a line: equal slopes; an unknown: an equation in the slope.
How to do it
Method
In a line: equal slopes; an unknown: an equation in the slope
k − 36 − 2 = 12, k = 5
Another way
Walk it
4 × 12 = 2, 3 + 2 = 5
In full
Slopes settle three kinds of question. ① Whether three points lie in a line: if A, B and C are on one line, AB's slope equals BC's (the two segments share B, so equal slopes mean no turn). A(1, 1), B(3, 5), C(6, 11): AB's slope is 42 = 2 and BC's 63 = 2, so they are in a line. ② An unknown coordinate from a slope: the line through A(2, 3) and B(6, k) has slope 12: k − 36 − 2 = 12, k − 3 = 2, k = 5. ③ How steep something really is: a ramp rising 1 m over 12 m across has slope 112, which is what a sign saying "1 : 12" means. Or walk it: a slope of 12 rises 12 a square for every square right; from x = 2 to x = 6 is 4 squares, rising 2, so k = 3 + 2 = 5, the same answer.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A line through A(0, −4) and B(6, k) has slope 13. k =
Show the answer and working
Answer: −2
k − (−4)6 − (0) = 13, so k − (−4) = 13 × 6 = 2 and k = −2.
Example 2
A line passes through A(1, −1) and B(−2, 1). Which of these points is on it too?
- A. (−5, 3)
- B. (0, −2)
- C. (−5, −1)
- D. (1, 3)
Choose the right answer
Show the answer and working
Answer: A. (−5, 3)
From A to B each part changes x by −3 and y by 2; from B to (−5, 3) x changes by −3 and y by 2 too: the same slope, so the three are in a line.
Example 3
A(−3, −5), B(−1, −3) and C(0, k) lie on one straight line. k =
Show the answer and working
Answer: −2
AB's slope is 11. BC's must be 11 too: k − (−3)0 − (−1) = 11, so k = −2.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A line through A(2, −3) and B(6, k) has slope 14. k =
Answer:
2.A(−1, −1), B(1, −2) and C(5, −4) lie on one straight line.
True or false?
Answer:
3.A line through A(−5, −4) and B(h, −6) has slope 13. h =
Answer:
4.A line passes through A(3, 6) and B(5, 9). Which of these points is on it too?
- A. (8, 11)
- B. (7, 6)
- C. (7, 12)
- D. (3, 12)
Choose the right answer
Answer:
5.A(3, 1), B(4, 4) and C(5, k) lie on one straight line. k =
Answer:
All the answers
- 1. −2
- 2. True
- 3. −11
- 4. C. (7, 12)
- 5. 7
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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