If the mid-point of the line segment joining the points $A(3, 4)$ and $B(k, 6)$ is $P(x, y)$ and $x + y - 10 = 0$, find the value of $k$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:28 · grounding rag
Model Answer
Mid-point of A(3, 4) and B(k, 6):
Using the mid-point formula:
$$P(x, y) = \left(\frac{3+k}{2},\ \frac{4+6}{2}\right) = \left(\frac{3+k}{2},\ 5\right)$$
So, $x = \dfrac{3+k}{2}$ and $y = 5$.
Applying the condition $x + y - 10 = 0$:
$$\frac{3+k}{2} + 5 - 10 = 0$$
$$\frac{3+k}{2} = 5$$
$$3 + k = 10$$
$$k = 7$$
Source: Chapter 7, Section 7.4 (Mid-point Formula)
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Explanation
- Examiners expect you to clearly apply the mid-point formula first, then substitute into the given condition — these are the two steps that earn marks.
- Write $y = 5$ explicitly; don't skip it, as it shows you found both coordinates.
- The condition $x + y - 10 = 0$ gives one equation in one unknown ($k$), making it straightforward to solve.
- Award split: ~1 mark for mid-point coordinates, ~1 mark for substitution into the condition, ~1 mark for correct value of $k$.