In the Desmos Geometry tool, you can translate, reflect, rotate, and dilate geometric objects. After defining a transformation, you can reuse it from your list of ‘Recent Transformations’ or even build composite transformations from the expression list. The descriptions of each transformation type below will help you get started, or check out the additional topics to dive deeper!

## Translate

To translate objects, first select them using the select tool. Once you’ve selected all of the objects you want to translate, click the ‘Transform’ button that appears in the Geometry toolbar and follow the prompts to choose a start and end point or a vector to translate by. You will translate all of the selected objects by the chosen vector. To try the example translation, click on the image to the right.

**Quick Tip**: To select many items at once, hold shift and drag to box-select.

## Reflect

To reflect objects, first define the line of reflection. This can be a segment, ray, or line. Next, select the objects you want to reflect with the select tool and click the ‘Transform’ button from the toolbar. Follow the reflection tool prompt to choose the line of reflection. You will reflect all of the selected objects over the same line. To try the example reflection, click on the image to the right.

## Dilate

To dilate (scale) objects, start by selecting them with the select tool and then clicking the ‘Transform’ button from the toolbar. Follow the prompts to first choose the center of dilation. This can be an existing point or a new point defined by where you click on the graph paper. Then, enter the scale factor in the input box. You will dilate all of the selected objects with respect to the chosen center. To try the example dilation, click on the image to the right.

**Quick Tip:** You can also use a variable or expression as the scale factor by entering them in the input box and then clicking ‘go.’ If you use a variable, this will automatically add a slider to the expression list.

## Rotate

The last transformation is a rotation. To rotate objects, first select the objects you want to rotate with the select tool. Then, click the ‘Transform’ button from the toolbar and select ‘Rotate.’ Follow the prompts to choose a point to define the center of rotation. Then, construct or choose a directed angle to rotate by. You can also type the rotation angle in the input box (both counterclockwise and clockwise angles are allowed). You will rotate all of the selected objects by the given angle about the center of rotation. To try the example rotation, click on the image to the right.

**Quick Tip:** You can also use a variable or expression as the angle by entering them in the input box and then clicking ‘go.’ If you use a variable, this will automatically add a slider to the expression list.

## More About Transformations

### Reuse Existing Transformations

After you apply a transformation, the image of your object stays selected and your transformation is added to a list of ‘Recent Transformations’ in the toolbar. This means you can reuse previously defined transformations! You can apply existing transformations to different objects, or apply them to the same object multiple times.

### Defining a Transformation from the Expression List

You can define transformations as functions in the expression list, using the commands ‘translate,’ ‘reflect,’ ‘dilate,’ and ‘rotate.’ For example, the function \(R(x)=\)rotate\((x, (0,0), 90)\) applies a \(90\) degree counterclockwise rotation around the point \((0,0)\) to any object \(x\). To apply the transformation \(R(x)\) to a geometry object on the graph paper, type R(), position your cursor inside the parentheses, and then hold shift and click the geometry object. Check out this rotation example here.

### Composite Transformations

You can also define transformations as functions from the expression list. When you do this, you can reuse the transformation function or even build a composite transformation. To see an example composition, visit the Connections Between Geometry and Algebra help center article.

## Learn More

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