Free printable resources · Original practice
Apply one named transformation to every point, preserve vertex order and compare the original and image on the same coordinate plane.
Free graphing transformations worksheets with translation, rotation, reflection and dilation cases on coordinate grids. Letter/A4 PDF and editable SVG answers included.
Choose Letter or A4, then download a student sheet first. Keep the separate reference or answer file closed until you have attempted the case. Every worksheet, SVG and answer is free without signup.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Complete free PDF pack.
| Case | Given information | Task |
|---|---|---|
| TR01 | Vertices in order: A(1, 1); B(4, 1); C(4, 3). | Translate by (2, 2). Draw and label the image. Check the coordinates of every vertex. |
| TR02 | Vertices in order: A(-2, 1); B(1, 1); C(0, 4). | Rotate 90 degrees counterclockwise about (0, 0). Draw and label the image. Check the coordinates of every vertex. |
| TR03 | Vertices in order: A(-3, -1); B(-1, -1); C(-2, 2). | Reflect across y = x. Draw and label the image. Check the coordinates of every vertex. |
| TR04 | Vertices in order: A(1, 1); B(3, 1); C(2, 4). | Dilate about (0, 0) by scale factor 2. Draw and label the image. State the length and area ratios. |
Use the coordinates, dimensions and rules given for each case. Distances and areas are measured in grid units.
TR01: Translate a triangle
| Step | Worked result |
|---|---|
| 1 | Translate by (2, 2). |
| 2 | Image vertices: A(3, 3); B(6, 3); C(6, 5). |
The first case shows the checking pattern. Individual answer files contain the completed graph or net and result for every case.
For a translation by (2, 2), add 2 to each x-coordinate and 2 to each y-coordinate. Starting with A(1, 1), B(4, 1), C(4, 3), the image vertices are A(3, 3), B(6, 3), C(6, 5). Plot these points in order and connect them to form the image. Verify each coordinate by counting grid units from the original point: move right 2 and up 2. The original area is 3 square units and the image area is also 3 square units, because translation preserves size and shape. Label all vertices clearly.
A 90-degree counterclockwise rotation about the origin maps (x, y) to (-y, x). Apply this to A(-2, 1): (-1, -2). For B(1, 1): (-1, 1). For C(0, 4): (-4, 0). Plot A(-1, -2), B(-1, 1), C(-4, 0) and connect them in order. Check that each image point is the same distance from the origin as its original. The original area is 9/2 square units and the image area is also 9/2 square units, since rotation preserves area. Label the image vertices with primes if needed.
A dilation about the origin by scale factor 2 multiplies each coordinate by 2. For A(1, 1), the image is A(2, 2). For B(3, 1), it is B(6, 2). For C(2, 4), it is C(4, 8). Plot and connect these points. The length ratio is 2, meaning each side of the image is twice the corresponding side of the original. The area ratio is the square of the scale factor, so it is 4. The original area is 3 square units and the image area is 12 square units. State both ratios clearly.
| Watch for | Check instead |
|---|---|
| When translating by (2, 2), adding 2 only to the x-coordinates and leaving y-coordinates unchanged. | Add 2 to both coordinates: (x, y) becomes (x+2, y+2). For example, A(1, 1) becomes A(3, 3), not A(3, 1). |
| For a 90-degree counterclockwise rotation, using the rule (x, y) -> (y, -x) instead of (-y, x). | The correct rule is (x, y) -> (-y, x). Check with A(-2, 1): (-1, -2), not (1, 2). |
| After a dilation by scale factor 2, stating the area ratio is 2 instead of 4. | The area ratio is the square of the scale factor: 2^2 = 4. The original area 3 becomes 12 square units, so the ratio is 12/3 = 4. |
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Verify that each image point is the same distance from the center of rotation as its original point. For a 90-degree counterclockwise rotation about the origin, also confirm that the angle between the original and image position vectors is 90 degrees.
A dilation by scale factor k multiplies every length by k. Area depends on two dimensions, so it is multiplied by k times k, which is k^2. For k=2, area becomes 4 times larger.
Translations, rotations, and reflections preserve distances and angles, so the image has the same area and perimeter as the original. Dilations preserve shape but change lengths by the scale factor and area by the square of the scale factor.
Yes. The individual student, reference and answer files plus collection packs are free without signup. Credit Zumola under CC BY 4.0 when sharing or adapting.
The diagrams and exercises on this page are original Zumola resources.
Try four more original cases. Every reference drawing, student worksheet and worked answer is free, with Letter and A4 PDFs and editable SVG files.
Additional original practice. Free Letter and A4 worksheets, drawings and answers.
Additional original practice. Free Letter and A4 worksheets, drawings and answers.
Additional original practice. Free Letter and A4 worksheets, drawings and answers.
Additional original practice. Free Letter and A4 worksheets, drawings and answers.
Additional original practice. Free Letter and A4 worksheets, drawings and answers.
To translate a figure, add the horizontal shift to every x-coordinate and the vertical shift to every y-coordinate. For example, given A(0, 2), B(3, 2), C(3, 4) and translation (2, 2), compute A'(0+2, 2+2) = (2, 4), B'(3+2, 2+2) = (5, 4), C'(3+2, 4+2) = (5, 6). Plot and label each image vertex. Check that corresponding sides remain parallel and equal in length. The area stays 3 square units. Always verify every vertex coordinate after drawing.
Vertices in order: A(0, 2); B(3, 2); C(3, 4).
The answer PDFs include the completed drawings and checks for all four cases. Free original resources, CC BY 4.0. No signup required.
A composite transformation is a chain. Treat each named rule as a separate pass over the vertex list. After the first pass, you have a real set of coordinates you can check before continuing. Combining two rules mentally invites sign errors, especially when a reflection and a translation both change the same coordinate.
Write the vertex order once at the top of your work, for example A, B, C. Use primes to mark stages: A, B, C for the original; A′, B′, C′ after the first rule; A″, B″, C″ after the second. This notation keeps the correspondence obvious and makes it easy to spot a vertex that was skipped.
Translations, rotations and reflections are rigid: distances and angles stay the same, so the image is congruent to the original. A dilation changes lengths by the scale factor while angles stay fixed, so the image is similar but not congruent unless the factor is 1. In a chain, note which stage is rigid and which is not, because that tells you what to verify at the end.
If the chain includes a dilation, the final side lengths equal the original side lengths multiplied by the scale factor, regardless of where the dilation sits in the order. Angles, however, are preserved throughout. Use these facts as independent checks rather than trusting the drawing alone.
Composite images often overlap the original, which makes vertex labels hard to read. Use a consistent convention: solid lines for the original, dashed for the intermediate, and a distinct style for the final image. Place labels slightly outside each vertex so they do not sit on a grid intersection shared by two figures.
If two vertices land on the same point, note it rather than hiding it. A shared vertex is a legitimate result of the transformation and is worth recording, because it shows the chain was applied correctly.
Triangle with vertices A(1, 1), B(4, 1), C(1, 3). First translate by (x, y) → (x + 2, y + 1), then reflect across the x-axis using (x, y) → (x, −y). Find the final vertices.
After translation: A′(3, 2), B′(6, 2), C′(3, 4). After reflection across the x-axis: A″(3, −2), B″(6, −2), C″(3, −4). Check: original side AB has length 3 and final A″B″ has length 3, consistent with two rigid moves. Original angle at A is 90 degrees and the final angle at A″ is also 90 degrees.
Often yes. Translation and reflection generally do not commute, so apply the rules in the stated order and record each stage.
Apply the coordinate rules arithmetically to each vertex in sequence and compare the resulting coordinates with your plotted points.
Angles are preserved and the shape remains similar; side lengths scale by the dilation factor, so congruence holds only when that factor is 1.
