7.02 Transformations and congruence
Rigid transformations
There are three special kinds of transformations that we will quickly review.
Translations slide objects, without changing their orientation.
Reflections flip objects across a line:
Rotations move an object around a central point by some angle.
What makes these three kinds of transformations special is that the original shape and the transformed shape have the same properties:
They have the same area
Every side length stays the same
Every internal angle stays the same
For this reason these three transformations are sometimes called rigid transformations. You can think of them as treating the shape as though it was made out of a rigid material, such as metal or hard plastic, with no stretching or squishing allowed.
Examples
Example 1
Which diagram shows two triangles that are reflections of one another?
Translations slide shapes around. Reflections flip shapes across a line. Rotations rotate shapes around a point. These rigid transformations preserve the area, side lengths, and internal angles of the shape.
Reflections, translations and rotations can be thought of as happening to the individual points of a shape.
If a shape is reflected but remains unchanged, then that line is an axis of symmetry.
Congruence
We say two shapes X and Y are congruent if we can use some combination of translations, reflections and rotations to transform one shape into the other. All of these shapes are congruent to each other:
We use the symbol \equiv to express this relationship, so we read A \equiv B as 'A is congruent to B'.
Examples
Example 2
The diagram below shows two triangles that are translations of one another:
Which of the following angles has the same size as \angle CBA?
Example 3
\triangle ABC is reflected along the dotted line and its image \triangle XYZ is produced.
Find the length of each side of \triangle XYZ.
\triangle XYZ is:
Two shapes A and B are congruent if we can use some combination of rigid transformations to transform one into the other.
We use the symbol \equiv to express this relationship, so we read A \equiv B as 'A is congruent to B'.
A line of symmetry is a line that when a shape is reflected across it, it remains unchanged.
Bisectors
A bisector is a line that cuts something into two equal halves. An angle bisector divides an angle into two angles of equal size, and a side bisector divides a side into two segments of equal length.
By contrast, a scalene triangle never has an axis of symmetry.
Examples
Example 4
Consider \triangle PQR with angle bisector PX of \angle QPR and QR=12:
What type of triangle is this?
Which of the following is true?
Is the line through P and X a line of symmetry for \triangle PQR?
A bisector is a line that cuts something into two equal halves. An angle bisector divides an angle into two angles of equal size, and a side bisector divides a side into two segments of equal length.
In an isosceles triangle, the line through a vertex bisecting the opposite side is also an angle bisector. This line is an axis of symmetry for the triangle, and meets the base at right angles.