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12.07 Transformations and congruence

Lesson

Rigid transformations

There are three special kinds of transformations that we will quickly review.

Translations slide objects, without changing their orientation.

This image shows a shape translated by 5 units to the right and 5 units down. Ask your teacher for more information.
This shape has been translated by 5 units to the right and 5 units down.
This image shows a shape translated 5 units to the right and 5 units down. Ask your teacher for more information.
Each point is translated 5 units to the right and 5 units down along with it.

Reflections flip objects across a line:

This image shows a reflection of a shape. Ask your teacher for more information.
A shape reflected about a line.
This image shows that every point of the shape is the same distance from the reflecting line, but on the opposite side.
Every point is the same distance from the reflecting line, on the opposite side.
This image shows a shape with the reflecting line crossing through it. Ask your teacher for more information.
The reflecting line may cross through an object.
This image shows that every point of the shape is the same distance from the reflecting line, but on the opposite side.
Points that lie on the reflecting line stay on the line.
This image shows three shapes divided through a line reflecting each side. Ask your teacher for more information.

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.

This image shows a rotation of a shape 90 degrees around a point. Ask your teacher for more information.
This shape has been rotated 90\degree clockwise around the point A.
This image shows a rotation of a polygon around a point at 90 degrees. Ask your teacher for more information.
Each point will stay the same distance from the central point 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.

Examples

Example 1

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

A
This image shows two triangles that are reflections of one another
B
This image shows two triangles that are rotations of one another
C
This image shows two triangles that are translations of one another.
D
This image shows two triangles that are rotations of one another.
Worked Solution
Create a strategy

Choose the option that shows two shapes that are mirror-images of each other.

Apply the idea

Option A is the correct answer since the two triangles are mirror-images of each other.

Idea summary

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:

This image shows quadrilaterals that are congruent to each other. Ask your teacher for more information.

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:

Triangle PQR is a translation of triangle ABC and vice versa.

Which of the following angles has the same size as \angle CBA?

A
\angle QPR
B
\angle RQP
C
\angle PRQ
Worked Solution
Create a strategy

Look for the angle in \triangle PQR that is also opposite the side with 3 markings.

Apply the idea

\angle CBA is the ange at point B. This angle is opposite the side with 3 markings.

In \triangle PQR point Q is opposite the side with 3 markings.

So \angle RQP has the same size as \angle CBA. Option B is the correct answer.

Example 3

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

Triangle XYZ is a reflection of triangle ABC with side lengths 7, 8 and 9. Ask your teacher for more information.
a

Find the length of each side of \triangle XYZ.

Worked Solution
Create a strategy

Identify the corresponding sides.

Apply the idea

CB corresponds to XY.

BA corresponds to YZ.

CA corresponds to XZ.

So we have the length of each side:

XY=8 cm

YZ=7 cm

XZ=9 cm

b

\triangle XYZ is:

A
congruent to \triangle ABC
B
an enlargement of \triangle ABC
Worked Solution
Create a strategy

A reflection of a figure is also known as a flip, and does not affect size.

Apply the idea

Since \triangle ABC is reflected along the dotted line and produced the image \triangle XYZ, then have the same area, every side length stays the same, and every internal angle stays the same.

So \triangle XYZ is congruent to \triangle ABC.

So option A is the correct answer.

Idea summary

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.

This image shows an angle with an angle bisector and a segment with side bisector.Ask your teacher for more information.
This image shows an isosceles triangle with an axis of symmetry. Ask your teacher for more information.

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.

This image shows shows an isosceles triangle divided into two triangles. Ask your teacher for more information.

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

By contrast, a scalene triangle never has an axis of symmetry.

This image shows shows a scalene triangle with its angle bisectors and side bisectors. Ask your teacher for more information.

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.

Examples

Example 4

Consider \triangle PQR with angle bisector PX of \angle QPR and QR=12:

Triangle P Q R with an angle bisector at P. Ask your teacher for more information.
a

What type of triangle is this?

Worked Solution
Create a strategy

Compare the side lengths.

Apply the idea

The side lengths are 10, \, 12 and 14, which are all unequal. So it is a scalene triangle.

b

Which of the following is true?

A
QX \gt RX
B
QX = RX
C
QX \lt RX
Worked Solution
Apply the idea

Angle bisectors bisect the opposite side. So since PX bisects \angle QPR it also bisects side QR.

So option B is correct QX=RX.

c

Is the line through P and X a line of symmetry for \triangle PQR?

Worked Solution
Apply the idea

No. \triangle PQX is not the mirror image of \triangle PXR since QP \neq PR.

Idea summary

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.

Outcomes

MA4-17MG

classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles

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