Points, distance and midpoint · Coordinate geometry

A point in the plane is written $(x, y)$ : $x$ is how far across from the origin and $y$ is how far up. Two formulas answer a lot of questions about two points $A(x_1, y_1)$ and $B(x_2, y_2)$ .

The distance $AB$ is the hypotenuse of a right-angled triangle whose other sides are the difference in $x$ and the difference in $y$ . So by Pythagoras:

AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

The midpoint $M$ of $AB$ is halfway along. Its coordinates are the averages of the coordinates of $A$ and $B$ :

M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)

DistanceAB² = (x₂ − x₁)² + (y₂ − y₁)²

MidpointM = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Try it

Two pointsDrag the gold points

6across, x₂ − x₁4up, y₂ − y₁√52 = 2√13 ≈ 7.21AB = √(across² + up²)

The distance is the hypotenuse of a right-angled triangle: 6 across and 4 up. By Pythagoras, AB = √(6² + 4²) = √52. Squaring makes the signs not matter.

Drag $A$ and $B$ . In “Distance” the triangle under $AB$ shows the two differences; however you place the points, $AB^2$ is the sum of their squares. In “Midpoint”, $M$ always sits at the average of the two $x$ ‘s and the two $y$ ‘s.

Working backwards

Often the midpoint is given and one end is missing. Write the midpoint formula for each coordinate and solve. A quick way: the missing end is as far beyond $M$ as the given end is before it, so $x_2 = 2x_M - x_1$ .

When a distance is given and a coordinate is unknown, put the unknown into the distance formula and solve. Squaring gives two answers, and often both are allowed.

Worked example · WAEC 2021

WAEC 2021 · Paper 2 · Q12 (b)

The coordinates of two points $P$ and $Q$ in a plane are $(7, 3)$ and $(5, x)$ respectively, where $x$ is a real number. If $|PQ| = \sqrt{29}$ units, find the values of $x$ .

Write the distance formula
$|PQ|^2 = (7 - 5)^2 + (3 - x)^2 = 29$ .
Think first. Put P(7, 3) and Q(5, x) into the formula, squared.
Simplify
$4 + (3 - x)^2 = 29$ , so $(3 - x)^2 = 25$ .
Think first. What is (7 − 5)²? Move it across.
Take the square root
$3 - x = 5$ or $3 - x = -5$ , so $x = -2$ or $x = 8$ . Both points are $\sqrt{29}$ from $P$ .
Think first. How many numbers square to 25?

Your turn

WAEC 2021 · Paper 1 · Q41

Coordinate geometry

Let $\overline{XY}$ be a line segment with $X(-8, -12)$ and $Y(p, q)$ . If the midpoint of $\overline{XY}$ is $(-4, -2)$ , find the coordinates of $Y$ .

Worked solution (try it first)

The midpoint is the average of the ends, so each coordinate of $Y$ is twice the midpoint's minus $X$ 's.
$p = 2(-4) - (-8) = -8 + 8 = 0$ .
$q = 2(-2) - (-12) = -4 + 12 = 8$ .
So $Y = (0, 8)$ , option B.

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