Lesson 9.4 · 20 min

Coordinates on a Line Segment

Finding the midpoint, and the point a given fraction of the way along a line segment, from the coordinates of its ends.

In this lesson you will learn to…

Find the midpoint of a line segment
Find the point that is a given fraction of the way along a segment
Work from either end, including with negative coordinates
Use a quick sketch to check the answer is sensible

Why this matters

Computer graphics, map-making and GPS all rely on finding points part-way between two others. Game designers use exactly this to move a character smoothly from one position to another, a fraction of the way each frame.

Sketch first

Even without squared paper, a quick sketch of the two points and the right-angled triangle between them tells you:

whether to add or subtract each distance, and
roughly where the answer should be — so you can spot a silly mistake.

The midpoint is the special case

The midpoint is $\frac{1}{2}$ of the way along. Using the fraction method for A $(2, 1)$ and B $(8, 10)$ : half of 6 is 3 and half of 9 is 4.5, giving $(5, 5.5)$ . The formula gives the same: $\left(\frac{2 + 8}{2}, \frac{1 + 10}{2}\right) = (5, 5.5)$ .

Check. A point $\frac{1}{3}$ of the way from A should be closer to A than to B. If yours is not, you probably started from the wrong end.

Key rules and formulas

Midpoint
$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Add the $x$ -coordinates and halve; add the $y$ -coordinates and halve.
A fraction of the way along
Find the horizontal and vertical distances from the start to the end. Take that fraction of each, and add it to the starting point.
Which end you start from matters
$\frac{1}{4}$ of the way along AB from A is a different point from $\frac{1}{4}$ of the way from B. From B you move towards A, so the steps change sign.

Worked examples

Worked example 1

A third of the way along

A is

(2, 1)

and B is

(8, 10)

. Find the point T that is

\frac{1}{3}

of the way along AB from A.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

From each end

A is

(3, 1)

and B is

(11, 17)

. Find the point a

\frac{1}{4}

of the way along AB from A b

\frac{1}{4}

of the way along BA from B.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

With negative coordinates

G is

(- 4, 14)

and H is

(6, - 1)

. Find the point

\frac{3}{5}

of the way along GH from G.

Try it yourself first, then reveal the solution one step at a time.

Practice: coordinates on a line segment

8 questions

Eight questions. Where two boxes are shown, type the

x

-coordinate in the first and the

y

-coordinate in the second.

Question 1 of 8Fill the gapsFoundation
Find the midpoint of A $(2, 1)$ and B $(8, 10)$ .
( Gap x , Gap y )
Question 2 of 8Step by step
Find the point one step at a time.
A is $(3, 1)$ and B is $(6, 16)$ . Find the point $\frac{1}{3}$ of the way along AB from A.
1. Horizontal distance from A to B:
  Step 1 answer
Question 3 of 8Fill the gaps
A is $(3, 1)$ and B is $(11, 17)$ . Find the point $\frac{1}{4}$ of the way along BA from B.
( Gap x , Gap y )
Question 4 of 8Fill the gaps
C is $(- 5, 15)$ and D is $(8, 7)$ . Find the point $\frac{1}{4}$ of the way along CD from C.
( Gap x , Gap y )
Question 5 of 8Fill the gaps
F is $(- 4, - 2)$ and E is $(2, 7)$ . Find the point $\frac{2}{3}$ of the way along EF from E.
( Gap x , Gap y )
Question 6 of 8Fill the gaps
G is $(- 4, 14)$ and H is $(6, - 1)$ . Find the point $\frac{3}{5}$ of the way along GH from G.
( Gap x , Gap y )
Question 7 of 8Multiple choice
P is $\frac{1}{3}$ of the way along AB from A. Which is true?
Question 8 of 8Fill the gapsChallenge
A is $(2, 1)$ and B is $(20, - 14)$ . Find the point $\frac{1}{3}$ of the way along AB from A.
( Gap x , Gap y )

In this lesson you will learn to…

Why this matters

Sketch first

The midpoint is the special case

Key rules and formulas

Midpoint

A fraction of the way along

Which end you start from matters

Worked examples

Practice: coordinates on a line segment