Transformation properties and proofs FAQ

We often use rigid transformations and dilations in geometric proofs because they preserve certain properties. Rigid transformations—such as translations, rotations, and reflections—preserve the lengths of segments, the measures of angles, and the areas of shapes. Dilations, on the other hand, change the size of a shape, but they preserve the measures of angles, the proportions, and relationships between different parts of the shape.

Certain transformations preserve even more properties. For example, when we translate or dilate a figure, figures that were parallel before the transformation are still parallel after it. So, when we use rigid transformations and dilations in geometric proofs, we can be confident that certain properties will remain the same even when the shapes are moved or resized.

Try it yourself with our Proofs with transformations exercise.

A counterexample is a specific instance (usually a mathematical example) that disproves a general statement or rule. Counterexamples are often used to refine definitions, to show where they break down, and to help us determine how to make them stronger.

For example, we might want to define a rotation of a geometric figure as a transformation that turns every point in the figure the same number of degrees around a center point. But, if we think about it, we can come up with a counterexample that shows this definition isn't quite right. If some points turn the same number of degrees, but in a different direction, we don't get a rotation.

So, using this counterexample, we might revise our definition to say that a rotation is a transformation of a geometric figure that turns every point in the figure the same number of degrees and in the same direction around a center point.

Counterexamples are a powerful tool in mathematics and can help us write better, more accurate definitions.

Try it yourself with our Defining transformations exercise.

Symmetry is all about how a shape looks the same after being transformed in some way. A shape has rotational symmetry if it looks the same after being rotated around a point. It has reflective symmetry if it looks the same after being reflected across a line.

So, can a figure have one type of symmetry but not the other? Absolutely! For example, an isosceles trapezoid has reflective symmetry across its vertical line of symmetry, but it doesn't have rotational symmetry. On the flip side, a parallelogram that is not a rhombus has rotational symmetry (it looks the same after being rotated $180\mathrm{°}$‍), but it doesn't have reflective symmetry.

Try it yourself with our Reflective symmetry of 2D shapes exercise and our Rotational symmetry of 2D shapes exercise.

We can define transformations more precisely using mathematical notation.

For a translation, we specify the displacement vector that describes the direction and magnitude of the movement. For example, we could write $T_{⟨5,2⟩}$‍ to represent moving an object $5$‍ units to the right and $2$‍ units up. We can also write a mapping function for the same translation:

For a reflection, we specify the line of symmetry. For example, we could write reflecting over the $y$‍-axis as $r_{y-\text{axis}}$‍. We could also use the equation of the line of reflection, such as $r_{x=0}$‍. In this case, the $y$‍-coordinate stays the same, but the $x$‍-coordinate flips to the opposite side of the $y$‍-axis.

For a rotation, we specify the point around which we rotate, the angle, and the direction. For example, we can describe rotating around the origin by $90\mathrm{°}$‍ clockwise as $R_{(0,0),-90\mathrm{°}}$‍. We'll be using positive angles for counterclockwise rotations and negative angles for clockwise rotations. Rotations around the origin in multiples of $90\mathrm{°}$‍ have straightforward mappings. Here is a mapping function for the rotation.

Rotations by other angles involve trigonometry, so we'll save those for a later course.