Finding an Endpoint from the Midpoint

The midpoint is the average of the two endpoints, so to find a missing endpoint you run that backwards: each endpoint coordinate is $2 \times \text{midpoint} - \text{known endpoint}$.

The point $M(3, 4)$ is the midpoint of the line segment $AB$. Given that $A$ is the point $(1, 2)$, find the coordinates of $B$. [3]

The working

Step 1, use the midpoint relationship. If $M$ is the midpoint of $A(x_A, y_A)$ and $B(x_B, y_B)$: $M = \left(\frac{x_A + x_B}{2}, \ \frac{y_A + y_B}{2}\right)$

Step 2, solve for $B$ by rearranging each coordinate ($x_B = 2x_M - x_A$): $x_B = 2(3) - 1 = 5 \quad \text{(M1)}$ $y_B = 2(4) - 2 = 6 \quad \text{(A1)}$

So $B = (5, 6)$. (A1)

Check: the midpoint of $A(1, 2)$ and $B(5, 6)$ is $\left(\frac{1+5}{2}, \frac{2+6}{2}\right) = (3, 4) = M$ ✓.

Where the marks are won and lost

• The rule is $B = 2M - A$ component by component, double the midpoint, subtract the known end. A common slip is to subtract in the wrong order ($A - 2M$).
• Apply it to both coordinates separately.
• A quick check (does $M$ come out as the midpoint of $A$ and your $B$?) confirms the answer cheaply.

Common mistakes

• Computing $M - A$ instead of $2M - A$ (that gives the displacement, not the endpoint).
• Subtracting in the wrong order.
• Applying the rule to only one coordinate.

Full method: Gradient, Midpoint & Length notes. Topic home: Straight-Line Graphs pillar.