The diagram shows the points \(A\) and \(B\). The point \(C\) is such that \(B\) is the midpoint of \(AC\). Find the coordinates of \(C\).

If \(B\) is the midpoint of \(AC\), then \(B\) is the average of \(A\) and \(C\). Rearranging the midpoint relationship gives the rule for finding the far end from one end and the midpoint:

\(\dfrac{x_A + x_C}{2} = x_B\), so \(x_C = 2x_B - x_A\), and likewise \(y_C = 2y_B - y_A\)

Reading from the diagram, \(A\) is \((-3, -2)\) and \(B\) is \((2, 3)\):

\(x\)-coordinate of \(C = 2 \times 2 - (-3) = 4 + 3 = 7\) [M1]

\(y\)-coordinate of \(C = 2 \times 3 - (-2) = 6 + 2 = 8\) [M1]

\(C = (7, 8)\) [A1]

An equivalent and often clearer route is to treat the move from \(A\) to \(B\) as a step and repeat it: from \((-3, -2)\) to \((2, 3)\) is 5 across and 5 up, so from \(B\) the same step again reaches \((7, 8)\).

Check by taking the midpoint of \(A\) and \(C\): \(\left(\frac{-3+7}{2}, \frac{-2+8}{2}\right) = (2, 3)\), which is \(B\). The usual error is averaging \(A\) and \(B\), which finds the midpoint of \(AB\) rather than extending the line beyond \(B\); the wording "\(B\) is the midpoint of \(AC\)" places \(B\) in the middle, so \(C\) must lie on the far side of \(B\) from \(A\).