Free printable resources · Original practice
Multiply both coordinates by the supplied scale factor, keep the origin fixed and compare perimeter and area changes.
Free dilation worksheets with integer and fractional scale factors. Plot the image, compare side lengths and use separate Letter/A4 answer files.
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 |
|---|---|---|
| DI01 | Vertices in order: A(1, 1); B(3, 1); C(1, 4). | Dilate about (0, 0) by scale factor 2. Draw and label the image. State the length and area ratios. |
| DI02 | Vertices in order: A(-4, 2); B(4, 2); C(4, 6); D(-4, 6). | Dilate about (0, 0) by scale factor 1/2. Draw and label the image. State the length and area ratios. |
| DI03 | Vertices in order: A(-1, 1); B(1, 1); C(0, 3). | Dilate about (0, 0) by scale factor 3. Draw and label the image. State the length and area ratios. |
| DI04 | Vertices in order: A(0, 0); B(3, 0); C(6, 3); D(0, 6). | Dilate about (0, 0) by scale factor 2/3. 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.
DI01: Double the triangle
| Step | Worked result |
|---|---|
| 1 | Dilate about (0, 0) by scale factor 2. |
| 2 | Image vertices: A(2, 2); B(6, 2); C(2, 8). |
| 3 | Length ratio = 2; area ratio = 4. Area: 3 to 12 square units. |
The first case shows the checking pattern. Individual answer files contain the completed graph or net and result for every case.
Start with triangle A(1, 1), B(3, 1), C(1, 4). To dilate about (0, 0) by scale factor 2, multiply each coordinate by 2: A becomes (2, 2), B becomes (6, 2), C becomes (2, 8). Plot these image points and connect them in the same order. The original area is 3 square units; the image area is 12 square units. The length ratio is 2, and the area ratio is 4. Check that each side length doubles and that the image area equals the original area times the square of the scale factor.
For rectangle A(-4, 2), B(4, 2), C(4, 6), D(-4, 6), dilate about (0, 0) by scale factor 1/2. Multiply each coordinate by 1/2: A(-2, 1), B(2, 1), C(2, 3), D(-2, 3). Plot and label the image. The original area is 32 square units; the image area is 8 square units. The length ratio is 1/2, and the area ratio is 1/4. Verify that each side length is halved and that the image area equals 32 times (1/2)^2, which is 8.
Take triangle A(-1, 1), B(1, 1), C(0, 3). Dilate about (0, 0) by scale factor 3 by multiplying each coordinate by 3: A(-3, 3), B(3, 3), C(0, 9). Plot the image and connect the vertices in order. The original area is 2 square units; the image area is 18 square units. The length ratio is 3, and the area ratio is 9. Confirm that each side length triples and that the image area equals the original area times 3 squared, or 9.
| Watch for | Check instead |
|---|---|
| Multiplying only one coordinate of each vertex by the scale factor. | Multiply both the x- and y-coordinate of every vertex by the scale factor. For example, dilating A(1, 1) by 2 gives (2, 2), not (2, 1). |
| Using the scale factor itself as the area ratio. | The area ratio is the square of the scale factor. For scale factor 2, the area ratio is 4; for 1/2, it is 1/4; for 3, it is 9; for 2/3, it is 4/9. |
| Connecting image vertices in a different order than the original. | Keep the same vertex order when plotting and connecting. For A(1, 1), B(3, 1), C(1, 4) dilated by 2, connect (2, 2) to (6, 2) to (2, 8) and back to (2, 2). |
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Multiply each coordinate of every vertex by the scale factor. For example, dilating A(0, 0), B(3, 0), C(6, 3), D(0, 6) by 2/3 gives A(0, 0), B(2, 0), C(4, 2), D(0, 4).
The length ratio equals the scale factor. The area ratio equals the square of the scale factor. For scale factor 2/3, the length ratio is 2/3 and the area ratio is 4/9; the area changes from 45/2 to 10 square units.
Verify that each image vertex is the original vertex multiplied by the scale factor, that side lengths scale by the same factor, and that the image area equals the original area times the square of the scale factor. For scale factor 3, area 2 becomes 18.
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The diagrams and exercises on this page are original Zumola resources.
