Geometry and measurement
Translations, Reflections and Rotations
Practise moving points and triangles with clear translation, reflection and rotation instructions, including worked coordinate answers.
Grades 6–8 practice · 6 questions · Free to print · Answers included
Make sense of the method
A translation slides a shape, a reflection flips it across a line, and a rotation turns it around a centre. These moves preserve lengths and angles.
This introductory set stays in the first quadrant. Reflection lines and rotation centres are named explicitly; broader Cartesian-plane work includes negative coordinates and more combinations.
- Identify the exact slide, mirror line or centre and turn direction.
- Transform every vertex using the same rule.
- Plot and label the image points, then join them in the original order.
- Check corresponding lengths. If there are two moves, perform them in the stated order.
A worked example
Translate T(2, 3) by 4 units right and 1 unit up.
| Vertical y / horizontal x | x 0 | x 1 | x 2 | x 3 | x 4 | x 5 | x 6 | x 7 | x 8 | x 9 | x 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| y 10 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 9 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 8 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 7 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 6 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 5 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 4 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 3 | Empty | Empty | T | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 2 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 1 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 0 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
- Add 4 to the x-coordinate: 2 + 4 = 6.
- Add 1 to the y-coordinate: 3 + 1 = 4.
- The image is T′(6, 4).
T′ = (6, 4).
| Vertical y / horizontal x | x 0 | x 1 | x 2 | x 3 | x 4 | x 5 | x 6 | x 7 | x 8 | x 9 | x 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| y 10 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 9 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 8 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 7 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 6 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 5 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 4 | Empty | Empty | Empty | Empty | Empty | Empty | T′ | Empty | Empty | Empty | Empty |
| y 3 | Empty | Empty | T | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 2 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 1 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
| y 0 | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty | Empty |
Try the worksheet
Start with the first two questions, then build toward the final challenge. Show your method with calculations, drawings or a short explanation. These are practice questions with published answers, not a full-topic readiness check.
Build the foundation
1.Translate P(5, 6) by 3 units left and 2 units down. Plot the image.
Translate point P: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty P Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Show your work on paper.Build the foundation
2.Triangle A(1, 1), B(4, 1), C(1, 3) moves 2 units right and 3 units up. Find all image vertices and check AB and A′B′.
Original triangle vertices: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty C Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty A Empty Empty B Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Show your work on paper.Strengthen your method
3.Reflect P(2, 3) across the vertical line x = 5. Draw the mirror line and find P′.
Reflect across x = 5: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty P Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Show your work on paper.Strengthen your method
4.Rotate P(6, 4) 90° counterclockwise about O(4, 4). Find P′ and show the radius before and after the turn.
Quarter-turn about O: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty O Empty P Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Show your work on paper.Apply and explain
5.Rotate triangle A(2, 2), B(4, 2), C(2, 3) by 180° about O(5, 5). Find the image vertices.
Half-turn triangle: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty O Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty C Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty A Empty B Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Show your work on paper.Apply and explain
6.Start with P(1, 2). First translate 2 units right, then reflect across x = 5. Compare that result with doing the reflection first and the translation second. Do the two orders agree?
Two transformation orders: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 y 12 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 11 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty P Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Show your work on paper.
Worked answers
Try the questions first. Then compare the reasoning, not just the final result.
Show all six worked solutions
1. P′ = (2, 4).
- Left means subtract from x: 5 − 3 = 2.
- Down means subtract from y: 6 − 2 = 4.
- Plot P′(2, 4).
Answer: translated P: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty P Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty P′ Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty 2. A′(3, 4), B′(6, 4), C′(3, 6). Both AB and A′B′ are 3 units long.
- Apply (x + 2, y + 3) to every vertex.
- A becomes (3, 4), B becomes (6, 4), and C becomes (3, 6).
- AB = 4 − 1 = 3; A′B′ = 6 − 3 = 3. Translation preserves the length.
Answer: translated triangle vertices: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty C′ Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty A′ Empty Empty B′ Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty 3. P′ = (8, 3).
- P is 5 − 2 = 3 units left of the mirror line.
- Place P′ 3 units right of it: x = 5 + 3 = 8.
- The y-coordinate stays 3. The mirror line is halfway between the points.
Answer: reflected P: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty P Empty Empty Empty Empty Empty P′ Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty 4. P′ = (4, 6).
- Relative to O, P is 2 units to the right.
- A 90° counterclockwise turn sends that direction upward.
- Move 2 units above O to (4, 6). Both radii are 2 units.
Answer: quarter-turn: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty P′ Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty O Empty P Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty 5. A′(8, 8), B′(6, 8), C′(8, 7).
- A half-turn puts the centre halfway between each point and its image.
- For this centre, calculate x′ = 10 − x and y′ = 10 − y.
- A′ = (8, 8), B′ = (6, 8), C′ = (8, 7).
Answer: half-turn triangle: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty B′ Empty A′ Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty C′ Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty O Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty 6. Translate then reflect gives (7, 2). Reflect then translate gives (11, 2). They differ.
- First order: (1, 2) → (3, 2) → (10 − 3, 2) = (7, 2).
- Second order: (1, 2) → (10 − 1, 2) = (9, 2) → (11, 2).
- The order matters: the reflection reverses the horizontal direction relative to its fixed mirror line.
Answer: compare final positions: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 y 12 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 11 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty P Empty Empty Empty Empty Empty A Empty Empty Empty B Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty
Mistakes worth catching
Reflecting across an axis when another line was named.
Draw the stated mirror line first and compare perpendicular distances.
Rotating around the origin by default.
Measure the point’s position relative to the named centre.
Assuming two transformations can be swapped.
Find the intermediate image for each order before comparing final images.
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Skills covered
Geometry and measurement
- Translate, reflect and rotate points or shapes using stated rules.
- Explain how coordinates change; these examples stay in the first quadrant.
Choose this practice by the skills you need. Grade placement and strand names vary between school systems; one worksheet covers selected skills rather than every expectation in a topic.
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