Midpoint of an Interval
The midpoint of an interval is the point halfway between its end points. Its coordinates are the averages of the end points' coordinates: \(M\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\). In NSW Year 10 Advanced maths the midpoint formula is used forwards and backwards, with surds and pronumerals, and to prove properties of shapes (NESA outcome MA5‑LIN‑P‑01).
You will find midpoints, find a missing end point from the midpoint, reflect a point through another point, and solve for unknowns in the coordinates. You will find a circle's centre and radius from a diameter, test whether a quadrilateral is a parallelogram by checking its diagonals, and rebuild a triangle from the midpoints of its sides. A sprinkler placed where a lawn's diagonals cross shows the idea in a garden. It is part of Year 10 Advanced Straight Line Graphs.
Every question with a fully worked solution.
- Midpoint of an Interval - Video - How To Find The Midpoint Using The Midpoint Formula Watch
Theory
The midpoint of an interval is the point halfway between its endpoints. Its coordinates are the averages: \(M = \left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)\). This Year 10 Advanced maths page covers the midpoint formula with fractions and surds, working back to a missing endpoint, and using midpoints to prove properties of shapes.
The midpoint \(M\) of an interval \(AB\) is the point on \(AB\) that is the same distance from \(A\) and from \(B\). It bisects the interval, cutting it into two equal halves.
Its \(x\)-coordinate is halfway between the \(x\)-coordinates of the endpoints, and its \(y\)-coordinate is halfway between their \(y\)-coordinates. Halfway between two numbers is their average.
Midpoints answer questions about shapes: the diagonals of a parallelogram bisect each other, so they share a midpoint, and the centre of a circle is the midpoint of any diameter.
The midpoint of the interval joining \((x_1,\ y_1)\) and \((x_2,\ y_2)\):
Rearranged to find the other endpoint \(B\) from \(A\) and \(M\):
How to find a midpoint
- Add the two \(x\)-coordinates and halve. Then do the same with the two \(y\)-coordinates.
- Simplify: add fractions with a common denominator, and simplify surds first so like surds combine.
- Write the answer as a point, \(M(x,\ y)\).
Working backwards and proving
- Missing endpoint or reflection in a point: set each average equal to the midpoint's coordinate and solve (Example 2).
- Unknown pronumerals: each coordinate gives one equation. Two unknowns need two equations, solved simultaneously (Example 4). If a point lies on a line such as \(y = 2x\), write it as \((a,\ 2a)\) first.
- Midpoint on an axis: set the matching coordinate to \(0\) and solve for the unknown.
- Parallelogram test: find the midpoints of both diagonals. Same point means the diagonals bisect each other, so it is a parallelogram (Example 3).
- Circles: the centre is the midpoint of a diameter; the radius is the distance from the centre to an endpoint. A point is on the circle if its distance from the centre equals the radius.
- Isosceles triangles: the line from the apex to the midpoint of the base is perpendicular to the base, so it is the height for the area.
- Midpoints of all three sides known: write \(x_1 + x_2\), \(x_2 + x_3\) and \(x_3 + x_1\) from them. Adding all three gives \(2(x_1 + x_2 + x_3)\); subtract each pair to get each \(x\). Repeat for \(y\).
(i) \(A(-3,\ 7)\) and \(B(6,\ -2)\)
(ii) \(P(\sqrt{18},\ 1)\) and \(Q(-\sqrt{2},\ -4)\).
Average the \(x\)-coordinates and average the \(y\)-coordinates.
(i)
| \(x\) | \(=\) | \(\dfrac{-3 + 6}{2} = \dfrac{3}{2}\) |
| \(y\) | \(=\) | \(\dfrac{7 + (-2)}{2} = \dfrac{5}{2}\) |
The midpoint is \(\boldsymbol{M\left(\dfrac{3}{2},\ \dfrac{5}{2}\right)}\). Leave fractions as fractions (or write \(1.5\) and \(2.5\)).
(ii) Simplify the surd first so that like surds can be added: \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).
| \(x\) | \(=\) | \(\dfrac{3\sqrt{2} + (-\sqrt{2})}{2} = \dfrac{2\sqrt{2}}{2} = \sqrt{2}\) |
| \(y\) | \(=\) | \(\dfrac{1 + (-4)}{2} = -\dfrac{3}{2}\) |
The midpoint is \(\boldsymbol{M\left(\sqrt{2},\ -\dfrac{3}{2}\right)}\).
Let \(B\) be \((x,\ y)\). Put each coordinate of the midpoint equal to the average, then solve.
| \(\dfrac{-4 + x}{2}\) | \(=\) | \(1\) |
| \(-4 + x\) | \(=\) | \(2\) |
| \(x\) | \(=\) | \(6\) |
| \(\dfrac{2 + y}{2}\) | \(=\) | \(-3\) |
| \(2 + y\) | \(=\) | \(-6\) |
| \(y\) | \(=\) | \(-8\) |
So \(\boldsymbol{B(6,\ -8)}\).
Check with steps: from \(A\) to \(M\) is \(5\) right and \(5\) down. The same step again from \(M\) lands on \((6,\ -8)\). This is also how to reflect a point through a centre point.
A quadrilateral is a parallelogram exactly when its diagonals bisect each other, that is, when \(AC\) and \(BD\) have the same midpoint.
| \(M_{AC}\) | \(=\) | \(\left(\dfrac{-3 + 5}{2},\ \dfrac{0 + 3}{2}\right) = \left(1,\ \dfrac{3}{2}\right)\) |
| \(M_{BD}\) | \(=\) | \(\left(\dfrac{2 + 0}{2},\ \dfrac{-1 + 4}{2}\right) = \left(1,\ \dfrac{3}{2}\right)\) |
The diagonals have the same midpoint, so they bisect each other: \(\boldsymbol{ABCD}\) is a parallelogram.
Each coordinate gives an equation. Multiply both by \(2\) to clear the fractions.
| \(\dfrac{a + 3b}{2} = 4\) | \(\Rightarrow\) | \(a + 3b = 8\) |
| \(\dfrac{2b + a + 1}{2} = 3\) | \(\Rightarrow\) | \(a + 2b = 5\) |
Subtract the second equation from the first, then substitute back.
| \(b\) | \(=\) | \(3\) |
| \(a\) | \(=\) | \(5 - 2 \times 3 = -1\) |
So \(\boldsymbol{a = -1}\), \(\boldsymbol{b = 3}\), giving \(\boldsymbol{P(-1,\ 6)}\) and \(\boldsymbol{Q(9,\ 0)}\). Check: \(\left(\dfrac{-1 + 9}{2},\ \dfrac{6 + 0}{2}\right) = (4,\ 3)\).
Common pitfalls
Frequently asked questions
What is the midpoint formula?
The midpoint of the interval joining \((x_1,\ y_1)\) and \((x_2,\ y_2)\) is \(\left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)\). You average the \(x\)-coordinates and average the \(y\)-coordinates.
How do you find an endpoint when you know the midpoint?
Let the unknown endpoint be \((x,\ y)\). Set \(\dfrac{x_1 + x}{2}\) equal to the midpoint's \(x\)-coordinate and solve, then do the same for \(y\). Or double the midpoint and subtract the known endpoint.
How do you find the centre of a circle from a diameter?
The centre is the midpoint of the diameter. Average the coordinates of the two ends of the diameter.
How do you prove a quadrilateral is a parallelogram with midpoints?
Find the midpoints of the two diagonals. If they are the same point, the diagonals bisect each other, and a quadrilateral whose diagonals bisect each other is a parallelogram.
Can a midpoint have fractions in it?
Yes. If the sum of the coordinates is odd, the midpoint has a half in it, such as \(\dfrac{3}{2}\). Give it as a fraction or a decimal.