Let’s compare measurements before and after translations, rotations, and reflections.
For each question, the unit is represented by the large tick marks with whole numbers.
Here is a grid showing triangle $ABC$ and two other triangles.
You can use a rigid transformation to take triangle $ABC$ to one of the other triangles.
Which one? Explain how you know.
Describe a rigid transformation that takes $ABC$ to the triangle you selected.
A square is made up of an L-shaped region and three transformations of the region. If the perimeter of the square is 40 units, what is the perimeter of each L-shaped region?
The transformations we’ve learned about so far, translations, rotations, reflections, and sequences of these motions, are all examples of rigid transformations. A rigid transformation is a move that doesn’t change measurements on any figure.
Earlier, we learned that a figure and its image have corresponding points. With a rigid transformation, figures like polygons also have corresponding sides and corresponding angles. These corresponding parts have the same measurements.
For example, triangle $EFD$ was made by reflecting triangle $ABC$ across a horizontal line, then translating. Corresponding sides have the same lengths, and corresponding angles have the same measures.
| measurements in triangle $ABC$ | corresponding measurements in image $EFD$ | |
|---|---|---|
| row 1 | $AB = 2.24$ | $EF = 2.24$ |
| row 2 | $BC = 2.83$ | $FD = 2.83$ |
| row 3 | $CA = 3.00$ | $DE = 3.00$ |
| row 4 | $m\angle ABC = 71.6^\circ$ | $m\angle EFD= 71.6^\circ$ |
| row 5 | $m\angle BCA = 45.0^\circ$ | $m\angle FDE= 45.0^\circ$ |
| row 6 | $m\angle CAB = 63.4^\circ$ | $m\angle DEF= 63.4^\circ$ |
If a part of the original figure matches up with a part of the copy, we call them corresponding parts. The part could be an angle, point, or side, and you can have corresponding angles, corresponding points, or corresponding sides.
If you have a distance between two points in the original figure, then the distance between the corresponding points in the copy is called the corresponding distance.
A rigid transformation is a sequence of translations, rotations, and reflections. If a rigid transformation is applied to a geometric figure, the resulting figure is called the image of the original figure under the transformation.
The diagram shows a rigid transformation consisting of a translation (from A to B) followed by a rotation (from B to C) followed by a reflection (from C to D). The last triangle is the image of the first triangle under this rigid transformation.
