Distance between points using Pythagoras | Year 9

01 / LEARN

The idea, step by step

Make a right triangle

Join the two points, then draw a horizontal step and a vertical step between them. These steps are perpendicular, so the joining segment is the hypotenuse of a right triangle. The leg lengths are the sizes of the coordinate differences.

Use Pythagoras' theorem to add the squares of those differences. Subtracting coordinates can produce a negative number, but its square is positive. Do not add the unsquared changes: that would measure a two-leg route rather than the direct segment.

d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

Exact or approximate?

If the sum of squares is a perfect square, the distance is an integer. Otherwise a square-root expression is an exact answer. Only convert it to a decimal when the question asks, and round once at the end.

For a horizontal or vertical segment, only one coordinate changes. Count that change directly or use the formula with the other difference equal to zero. In an application, include the unit and interpret the distance as a straight-line separation.

A common trap

Using $4 + 5 = 9$ for the direct distance in the second example.

Instead: Nine is the length of the horizontal-then-vertical route. The direct route is the hypotenuse, found using the square root of the sum of squares.

02 / WORKED EXAMPLES

See the thinking

Example 1

A coordinate triangle

Find the distance from $A (- 2, - 1)$ to $B (4, 7)$ .

The coordinate changes are $4 - (- 2) = 6$ and $7 - (- 1) = 8$ .
Pythagoras gives $d^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$ .
Take the non-negative square root: $d = 10$ .

10 units

Example 2

A non-integer distance

Find the distance between $(1, 2)$ and $(5, 7)$ , first exactly and then to 2 decimal places.

The changes are 4 and 5.
$d = \sqrt{4^{2} + 5^{2}} = \sqrt{41}$ .
$\sqrt{41} = 6.403 \dots$ , so the rounded distance is 6.40.

\sqrt{41}

units; approximately 6.40 units

Check your understanding

Find the distance between $(0, 0)$ and $(6, 8)$ .

Show answer

10 units

$d = \sqrt{36 + 64} = \sqrt{100} = 10$ .

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the distance between $(1, 2)$ and $(4, 6)$ .
2 marks
Find the distance between $(- 3, - 2)$ and $(5, 4)$ .
2 marks
Find the exact distance between $(0, 1)$ and $(3, 5)$ .
2 marks
Find the exact distance between $(2, - 1)$ and $(7, 1)$ , then round it to 1 decimal place.
3 marks
A map grid uses 100 m per coordinate unit. A gate is at $(1, 1)$ and a shelter at $(4, 5)$ . Find their straight-line separation.
3 marks
Points $A (- 2, 3)$ , $B (4, 3)$ and $C (4, - 5)$ form a triangle. Find its perimeter.
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the distance between $(1, 2)$ and $(4, 6)$ .
2 marks
The changes are 3 and 4.
$d = \sqrt{9 + 16} = 5$ .
5 units
Find the distance between $(- 3, - 2)$ and $(5, 4)$ .
2 marks
The changes are 8 and 6.
$d = \sqrt{64 + 36} = 10$ .
10 units
Find the exact distance between $(0, 1)$ and $(3, 5)$ .
2 marks
The changes are 3 and 4.
$d = \sqrt{3^{2} + 4^{2}} = 5$ .
5 units
Find the exact distance between $(2, - 1)$ and $(7, 1)$ , then round it to 1 decimal place.
3 marks
The changes are 5 and 2.
$d = \sqrt{25 + 4} = \sqrt{29} \approx 5.385$ .
To 1 decimal place this is 5.4.
$\sqrt{29}$ units; 5.4 units
A map grid uses 100 m per coordinate unit. A gate is at $(1, 1)$ and a shelter at $(4, 5)$ . Find their straight-line separation.
3 marks
The grid distance is $\sqrt{3^{2} + 4^{2}} = 5$ units.
Multiply by 100 m per unit.
500 m
Points $A (- 2, 3)$ , $B (4, 3)$ and $C (4, - 5)$ form a triangle. Find its perimeter.
4 marks
$A B = 6$ and $B C = 8$ .
$A C = \sqrt{6^{2} + 8^{2}} = 10$ .
The perimeter is $6 + 8 + 10 = 24$ .
24 units

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Gradient, midpoint and distance; algebraic and graphical straight-line relationships; interpreting models and testing parameters. Parallel and perpendicular lines are included as a natural school-course extension. All explanations and questions are original.

Official curriculum source ↗

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