Find the coordinates of the point C which lies on the line AB produced such that AC = 2BC, where coordinates of points A and B are $(-1, 7)$ and $(4, -3)$ respectively.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
Since C lies on line AB produced (beyond B) such that AC = 2BC, point C divides AB externally, with B between A and C.
AC = 2BC ⟹ AC – BC = BC ⟹ AB = BC, so B divides AC in ratio 1 : 1? No —
Since AC = 2BC and C is beyond B: AB + BC = AC = 2BC ⟹ AB = BC.
So B is the midpoint of AC. Let C = (x, y).
Using midpoint formula:
$$4 = \frac{-1 + x}{2} \Rightarrow x = 9$$
$$-3 = \frac{7 + y}{2} \Rightarrow y = -13$$
∴ Coordinates of C = (9, –13)
Source: Chapter 7, Section 7.3 (Section Formula / Midpoint Formula)
Explanation
- "AB produced" means C lies beyond B on the same line.
- AC = 2BC and C is on the far side of B, so AC = AB + BC = 2BC, giving AB = BC — B is the midpoint of AC.
- Apply midpoint formula: B = ((x_A + x_C)/2, (y_A + y_C)/2) and solve for C.
- Examiners award marks for correctly identifying B as midpoint and applying the formula accurately.