Rigid transformations
There are three special kinds of transformations that we will quickly review.
Translations slide objects, without changing their orientation. The shape below has been translated by $5$5 units to the right and $5$5 units down:
Notice that each point is translated $5$5 units to the right and $5$5 units down along with it:
Reflections flip objects across a line:
Every point of the object is the same distance from the reflecting line, but on the opposite side:
The reflecting line may cross through an object, like this:
As before, each point on the original object is the same distance from the reflecting line, but on the opposite side. Points that lie on the reflecting line stay on the line:
An object that looks exactly the same before and after a reflection have an axis of symmetry. Here are some examples:
Rotations move an object around a central point by some angle. The shape below has been rotated $90^\circ$90°clockwise around the point $A$A.
We can imagine this rotation happening to every point in the shape. Importantly, each point will stay the same distance from the central point $A$A:
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
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.
Practice questions
Question 1
Which diagram shows two triangles that are reflections of one another?
Two identical triangles with vertices marked with colored arcs are side by side. The triangles are the mirror image of each other. For the triangle on the left, the vertex on the top is marked with a red arc; the vertex on the right is marked with a blue arc; the vertex at the bottom is marked with a yellow arc. For the triangle on the right, the vertex on the left is marked with a blue arc; the vertex on top is marked with a red arc; the vertex at the bottom is marked with yellow arc.
A
Two triangles with vertices marked with colored arcs are side by side. The triangle on the right is identical to the triangle on the left but has different orientation. For the triangle on the left, the vertex on top is marked with a red arc; the vertex on the right is marked with a blue arc; the vertex at the bottom is marked with a yellow arc. For the triangle on the right, the vertex at the bottom is marked with a red arc; the vertex on left is marked with a blue arc; the vertex at the top is marked with a yellow arc.
B
Two identical triangles with vertices marked with colored arcs are side by side. The triangles are on the same position and orientation relative to each other. For both triangles: the vertex on the top is marked with a red arc; the vertex on the right is marked with a blue arc; the vertex at the bottom is marked with yellow arc. C
Two triangles with vertices marked with colored arcs are side by side. For the triangle on the left, the vertex on the top is marked with a red arc; the vertex on the right is marked with a blue arc; and the vertex at the bottom is marked with yellow arc. For the triangle on the right, the vertex on the top is marked with a blue arc; the vertex on right is marked with a red arc; the vertex at the bottom is marked with a yellow arc.
Both triangles have the same vertex at the bottom marked with a yellow arc but the vertices on top and on the right are switched.
D
Question 2
The diagram below shows two triangles that are translations of one another:
Two triangles, which are translations of one another, have their vertices marked with solid dots. The triangle on the left, $\triangle ABC$△ABC, has its vertices labeled as $A$A, $B$B and $C$C. The angle at vertex $A$A, $\angle BAC$∠BAC, is opposite side $BC$BC. The angle at vertex $B$B, $\angle ABC$∠ABC, is opposite side $AC$AC. The angle at vertex $C$C, $\angle ACB$∠ACB, is opposite side $AB$AB. The triangle on the right, $\triangle PQR$△PQR has vertices labeled as $P$P, $Q$Q and $R$R. The angle at vertex $P$P, $\angle QPR$∠QPR, is opposite side $QR$QR. The angle at vertex $Q$Q, $\angle PQR$∠PQR, is opposite side $PR$PR. The angle at vertex $R$R, $\angle PRQ$∠PRQ, is opposite side $PQ$PQ. Both side $AB$AB of $\triangle ABC$△ABC and side $PQ$PQ of $\triangle PQR$△PQR are marked with single tick marks. Both side $BC$BC of $\triangle ABC$△ABC and side $QR$QR of $\triangle PQR$△PQR are marked with double tick marks. Both side $AC$AC of $\triangle ABC$△ABC and side $PR$PR of $\triangle PQR$△PQR are marked with triple tick marks. Both $\angle ACB$∠ACB of $\triangle ABC$△ABC and $\angle PRQ$∠PRQ of $\triangle PQR$△PQR are opposite the sides marked with single tick marks. Both $\angle BAC$∠BAC of $\triangle ABC$△ABC and $\angle QPR$∠QPR of $\triangle PQR$△PQR are opposite the sides marked with double tick marks. Both $\angle ABC$∠ABC of $\triangle ABC$△ABC and $\angle PQR$∠PQR of $\triangle PQR$△PQR are opposite the sides marked with triple tick marks.
Which of the following angles has the same size as $\angle CBA$∠CBA?
$\angle QPR$∠QPR
A$\angle RQP$∠RQP
B$\angle PRQ$∠PRQ
C