Grade 8 · Grade 8: Transformations, Pythagoras and Volume · Transformations, Pythagoras and Volume
Transformations, Congruence and Similarity
Translate, reflect, rotate and dilate figures, and use transformations to define congruence and similarity.
Learning objectives
- Verify the properties of translations, reflections and rotations.
- Describe sequences of transformations mapping one figure onto another.
- Describe transformation effects using coordinates.
- Use angle facts and the AA criterion for similarity.
AERO Mathematics alignment
AERO.M8.GM.1
Verify experimentally the properties of rotations, reflections and translations of lines, line segments, angles and parallel lines.
AERO.M8.GM.2
Describe sequences of transformations that map one figure onto another, and understand congruence through rigid motions and similarity through dilations.
AERO.M8.GM.3
Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
AERO.M8.GM.4
Establish facts about the angle sum and exterior angle of triangles, angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
ENGAGE
Start here
A logo is slid, flipped, turned and then enlarged.
Think about it
Which of these four moves changes the size of the logo, and which keep it identical?
Hint: Compare lengths and angles before and after each move.
EXPLAIN
Rigid motions and dilations
| Transformation | What it does | Coordinate rule | Preserves |
|---|---|---|---|
| Translation | Slides | (x + a, y + b) | Length and angle |
| Reflection in y-axis | Flips | (−x, y) | Length and angle |
| Reflection in x-axis | Flips | (x, −y) | Length and angle |
| Rotation 180° about origin | Turns | (−x, −y) | Length and angle |
| Dilation, factor k | Resizes | (kx, ky) | Angle only |
Translations, reflections and rotations are rigid motions: they preserve length and angle, so the image is congruent to the original. A dilation preserves angles but multiplies lengths by k, so the image is similar but usually not congruent.
Angles with parallel lines
When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal and co-interior angles add to 180°. The angles of a triangle add to 180°, which follows directly from these facts, and an exterior angle equals the sum of the two opposite interior angles.
The AA criterion
If two angles of one triangle equal two angles of another, the triangles are similar. Their sides are then in a fixed ratio.
Remember
Congruent means same shape and same size. Similar means same shape, possibly different size.
INVESTIGATION
Try it yourself
Plot a triangle on grid paper, then apply a reflection, a rotation and a dilation of factor 2 in turn. Record the coordinates at each stage and state which properties changed.
Watch
Introduction to Transformations
Khan Academy · 7:22
Shows congruence and similarity as movements rather than lists of rules.
Before you watch: Which transformation changes size but not shape?
- Describe a sequence of transformations mapping one triangle onto its image.
- Why does a dilation preserve angle measures?
Interactive simulation · PhET
Area Model Algebra
Dilate a rectangle by scale factors of 2 and 3 and record lengths and areas.
While you explore
- Which properties stay the same under a dilation?
- Why is a dilation not a rigid motion?
Key vocabulary
- Dilation
- An enlargement or reduction from a centre point.
- Rigid motion
- A move that does not change size or shape.
Practice questions
0/1 correct
Level 1 · Criterion A
Which transformation produces a similar but not congruent figure?
MYP criterion tasks
Level 2 · Criterion C
A point (3, −5) is reflected in the y-axis then translated 2 units up. Give the final coordinates and explain each step.
Reflect & track
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