Topics
7. Geometry
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Middle Years

7.02 Transformations and congruence

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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
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.

 

Rigid transformations

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 $X$X and $Y$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 $A\equiv B$A≡B as "$A$A is congruent to $B$B".

Worked example

Are the two shapes below congruent?

Think: The two shapes are congruent if we can use some combination of reflections, translations and rotations to align the two shapes exactly.

Do: Find some combination of transformations for to align the two shapes.

If we reflect $X$X and then rotate it clockwise by $140^\circ$140° we will have the shape $Y$Y. So $X\equiv Y$X≡Y.

 

Reflect: Now when looking at the original shapes we also know which sides and angles correspond to each other.

 

Congruence

Two shapes $A$A and $B$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 $A\equiv B$A≡B as "$A$A is congruent to $B$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.

Exploration

An isosceles triangle always has an axis of symmetry - if we make a line through the vertex opposite the base and through the middle of the base, the line bisects the base. It meets the base at a right angle, and also bisects the angle at the top of the triangle:

The isosceles triangle can now been split in half forming two congruent triangles:

By contrast, a scalene triangle never has an axis of symmetry. The line through a vertex bisecting the opposite side is always different to the line through the vertex bisecting the angle there, and these lines never meet at right angles with the sides. Here is an example:

 

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.

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.

 

Practice questions

Question 1

Which diagram shows two triangles that are reflections of one another?

  1. 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:

  1. 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
Question 3

$\triangle ABC$△ABC is reflected along the dotted line and its image $XYZ$XYZ is produced.

  1. Find the length of each side:

    $XY=$XY=$\editable{}$ cm

    $YZ=$YZ=$\editable{}$ cm

    $XZ=$XZ=$\editable{}$ cm

  2. $\triangle XYZ$△XYZ is:

    congruent to $\triangle ABC$△ABC

    A

    an enlargement of $\triangle ABC$△ABC

    B

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