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# Pythagoras on a Coordinate Grid
- URL: https://www.esheets.io/pythagoras-on-a-coordinate-grid/
- Published: 2026-10-09T17:37:24.000Z
- Updated: 2026-10-09T17:37:24.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

Use Pythagoras to find straight-line distances between points on a coordinate grid. This GCSE worksheet builds from plotted first-quadrant segments to negative and decimal coordinates, triangle perimeters and a reverse problem with two possible answers.

Work through the questions, or review the [Topic guide](#topic-guide) for the right-triangle method and worked examples. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Find straight-line distances by turning horizontal and vertical coordinate changes into the perpendicular sides of a right-angled triangle.

Eight self-marking questions progress from scaffolded diagrams to calculations from coordinates, perimeter and a reverse-coordinate challenge.

#### What you’ll practise

- Reading horizontal and vertical changes across positive and negative axes
- Using Pythagoras and rounding non-integer distances to 1 decimal place
- Finding a triangle perimeter and both possible values of an unknown coordinate

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked examples.

## Pythagoras on a Coordinate Grid

Use the horizontal and vertical changes to form a right-angled triangle, then apply Pythagoras. Give decimal answers to 1 decimal place where requested.

Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

## Topic guide

### Distance as a right-angled triangle

To find the straight-line distance between two points, imagine drawing a horizontal line and a vertical line to make a right-angled triangle. The horizontal change and vertical change are its two shorter sides. The line joining the original points is the hypotenuse.

1. Find the horizontal change: **|x₂ − x₁|**.
2. Find the vertical change: **|y₂ − y₁|**.
3. Use **a² + b² = c²**, then take the positive square root.
4. Round only at the end if the question asks for a decimal answer.

### Worked example: crossing the axes

Find the distance from P = (−3, 4) to Q = (5, −2).

The horizontal change is |5 − (−3)| = 8\. The vertical change is |−2 − 4| = 6\. Notice that crossing zero makes the change larger; it does not make the distance negative.

Therefore PQ² = 8² + 6² = 64 + 36 = 100, so PQ = √100 = **10 units**.

Counting 8 squares across and then 6 squares down would give 14, but that is a route along two sides. The straight-line distance is the hypotenuse, so it is 10.

### Non-integer distances and rounding

If the squared length is not a square number, keep the square root on your calculator until the final step. For example, horizontal and vertical changes of 5 and 4 give √(5² + 4²) = √41 = 6.403…, which is **6.4 to 1 decimal place**.

### Reverse coordinates: why there can be two answers

Suppose A = (2, 3), B = (8, p) and AB = 10\. The horizontal change is 6, so the vertical change satisfies 6² + (p − 3)² = 10². This gives (p − 3)² = 64, so p − 3 = 8 or p − 3 = −8\. Therefore **p = 11 or p = −5**.

The two answers place B the same vertical distance above or below A. In compact form, the distance formula is √((x₂ − x₁)² + (y₂ − y₁)²), but it represents exactly the same right triangle.

### Common mistakes

- Do not add the horizontal and vertical changes: that counts a route around the triangle.
- Use brackets when subtracting a negative coordinate.
- Distances are never negative, even when coordinates are negative.
- For a perimeter, calculate all three side lengths before adding them.
- In a reverse problem, remember both the positive and negative square roots.