In the Desmos Geometry Tool, you can translate, reflect, rotate, and dilate geometric objects. After defining a transformation, you can reuse it or even build composite transformations from the expression list.
Translate
To translate objects, first make sure you have chosen Select from the toolbar. Then, select one or more objects you’d like to translate.
Click Transform and then Translate. Follow the prompts to choose a start and end point or a vector to translate by. You will translate all the selected objects by the chosen vector.
To select many items at once, hold Shift and click each object or 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.
Click Transform and then Reflect. Follow the prompt to choose the line of reflection. You will reflect all of the selected objects over the same line.
Dilate
To dilate (scale) objects, first select the objects.
Then, click Transform and then Dilate. 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 and press Go. You will dilate all of the selected objects with respect to the chosen center.
You can also use a variable or expression as the scale factor. If you use a variable, its slider will automatically appear in the expression list.
Rotate
The last transformation is a rotation. First select the objects you’d like to rotate. Then, click Transform and then 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.
You can use a variable or expression as the angle. 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 the Recent Transformations list that appears when you click Transform. You can apply existing transformations to different objects or apply them to the same object multiple times.
Define a Transformation from the Expression List
You can define transformations as functions in the expression list by typing Translate, Reflect, Dilate, or 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 \(\mathbf{R(0)}\), position your cursor inside the parentheses, and then hold Shift and click the geometry object. Try this rotation example.
Composite Transformations
When you define a transformation in the expression list, you can reuse the transformation function or even build a composite transformation. To see an example composition, visit the Connections Between Geometry and Algebra article.
Learn More
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