Use your method (the one used for M (−3, −4), A (0, 0) and G (6, 8)) to check if the points R (−5, −1), B (−2, −5) and C (4, −12) are on the same straight line. Now plot both sets of points and check your answers.

Step-by-step solution

Idea: The method: three points are on one straight line exactly when the two smaller distances between them add up to the largest. When the sum is even slightly bigger, the points form a (very thin) triangle.

1. RB = √((−2 − (−5))² + (−5 − (−1))²) = √(3² + (−4)²) = √25 = 5.½ mark
2. BC = √((4 − (−2))² + (−12 − (−5))²) = √(6² + (−7)²) = √85 ≈ 9.220.½ mark
3. RC = √((4 − (−5))² + (−12 − (−1))²) = √(9² + (−11)²) = √202 ≈ 14.213.½ mark
4. RB + BC = 5 + √85 ≈ 14.220, which is not equal to RC ≈ 14.213. (Exactly: (5 + √85)² = 110 + 10√85 and 10√85 = √8500 > √8464 = 92, so (5 + √85)² > 202 = RC².) So R, B, C are not on the same straight line.1 mark
5. Plotting both sets: M, A, G lie exactly on one line, as found before. R, B, C look almost in line, because the difference is only about 0.007 units, far too small to see. The shifts tell the same story: R to B is 3 across and 4 down, but B to C is 6 across and 7 down, not 8, so the path bends slightly at B. Calculation settles what the eye cannot.½ mark

Check: Shifts: R → B is (3, −4); for C to be on line RB, the shift B → C would have to be a multiple of (3, −4), such as (6, −8), which would lead to (4, −13). It is (6, −7), so C is 1 unit above that point, off the line ✓.

Answer to write in the exam

Common mistakes that cost marks

• Deciding from the plot alone. The points look collinear, but the distances show they are not.
• Rounding too early, e.g. √85 ≈ 9.2 and √202 ≈ 14.2, then saying 5 + 9.2 = 14.2 “so they are collinear”. Keep three decimal places, or compare squares exactly.
• Sign slips with negatives, e.g. −12 − (−1) = −13. It is −12 + 1 = −11.

How this can come in the exam

Try one yourself

Use the distance method to check whether (−3, 2), (0, 6) and (2, 9) lie on one straight line.

More questions like this

• Using the origin as one vertex, plot the vertices of:
• The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer.
• Use the connection you found to find the coordinates of B given that M (−7, 1) is the midpoint of A (3, −4) and B (x, y).
• Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find the coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, −2).
• (i) Given the points A (1, −8), B (−4, 7) and C (−7, −4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
  (ii) Given the points D (−5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.

All Coordinate geometry questions · All maths questions