📐 Grade 8 · Lesson 9 of 15
Translations, Rotations, Reflections & Congruence
Slide it, spin it, or flip it — none of these moves can stretch or squash a shape, so the copy is always congruent to the original.
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The big idea: A translation, a rotation, and a reflection are the three <strong>rigid motions</strong>: moves that carry a figure to a new position without changing its size or shape. Every length and every angle measure is preserved, so the moved figure is always <strong>congruent</strong> to the original. Two figures are congruent precisely when some sequence of rigid motions carries one exactly onto the other — that is the modern, motion-based definition of congruence.
Three ways to move a shape without changing it
In Grade 6 you reflected single points across an axis. Now the whole figure moves — every point of a triangle, a flag, a letter shape — and there are three basic moves that never distort it:
- A translation slides every point the same distance in the same direction.
- A rotation turns every point the same angle around a fixed centre point.
- A reflection flips every point across a line, called the line of reflection.
These three — together called rigid motions (or isometries) — share one defining property: they never stretch, shrink, or distort the figure they move.
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Rigid motions preserve length and angle
Every rigid motion keeps every distance between two points and every angle measure exactly the same as in the original figure. If a side was 5 cm long before the move, it is still 5 cm long after. If an angle was 40°, it is still 40°. Nothing about the figure's size or shape changes — only its position (and sometimes its orientation) does.
Congruence, redefined
Because rigid motions preserve length and angle, a figure and its image under a rigid motion always match up exactly — same side lengths, same angle measures. That match is exactly what congruent means.
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The motion definition of congruence
Two figures are congruent if and only if there is some sequence of translations, rotations, and reflections that carries one figure exactly onto the other. This replaces the older “same size and shape” description with something you can actually test: try to find the moves.
Coordinate rules you can use directly
On the coordinate plane, each rigid motion has a simple algebraic rule. Translating a point (x, y) by a units horizontally and b units vertically sends it to (x + a, y + b). Reflections and rotations about the origin follow their own fixed rules, shown below.
\[ (x, y) \;\xrightarrow{\text{translate by } (a,b)}\; (x+a,\; y+b) \]
A translation adds the same shift to every point's coordinates.
\[ (x,y)\xrightarrow{x\text{-axis}}(x,-y) \qquad (x,y)\xrightarrow{y\text{-axis}}(-x,y) \qquad (x,y)\xrightarrow{y=x}(y,x) \]
Reflection rules across the x-axis, the y-axis, and the line y = x.
\[ (x,y)\xrightarrow{90^{\circ}}(-y,x) \qquad (x,y)\xrightarrow{180^{\circ}}(-x,-y) \qquad (x,y)\xrightarrow{270^{\circ}}(y,-x) \]
Rotation about the origin, counterclockwise, by 90°, 180°, and 270°.
🎮 Rigid Motion Studio LIVE
Translate, rotate, or reflect a polygon and watch its side lengths and angle measures stay locked to the original — the hallmark of a rigid motion.
📝 Worked example: Triangle A has a vertex at (2, 3). Rotate this vertex 90° counterclockwise about the origin, then reflect the result across the y-axis. Find the final coordinates.
- Rotate 90°: \( (x,y)\to(-y,x) \), so (2, 3) \( \to (-3, 2) \).
- Reflect across the y-axis: \( (x,y)\to(-x,y) \), so (−3, 2) \( \to (3, 2) \).
- Both moves are rigid motions, so the final point is still the image of a congruent triangle — only its position and orientation changed.
✓ The final vertex is at <strong>(3, 2)</strong>.
📝 Worked example: Quadrilateral P has sides 4, 6, 4, 6 and angles 80°, 100°, 80°, 100°. It is translated 5 units right and 2 units up to form quadrilateral P′. What are the side lengths and angles of P′?
- A translation is a rigid motion, so it preserves every length and every angle measure.
- The sides of P are 4, 6, 4, 6 — the sides of P′ are the same: 4, 6, 4, 6.
- The angles of P are 80°, 100°, 80°, 100° — the angles of P′ are the same: 80°, 100°, 80°, 100°.
✓ P′ has sides 4, 6, 4, 6 and angles 80°, 100°, 80°, 100° — identical to P, because P and P′ are congruent.
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A reflection flips orientation — a rotation doesn't
A reflection reverses orientation: it turns a figure into its mirror image, like flipping a pancake. A rotation never does this — it just turns the figure in place, same “handedness” throughout. The trap: on a symmetric shape (like a square or an isosceles triangle), a 180° rotation can look exactly like a reflection from a distance, but it is not one. To tell them apart for certain, use an asymmetric figure (like a lopsided arrow or the letter F) — a reflection will always produce its mirror-flipped version, while a rotation never will, no matter the angle.
Check your understanding
1. Which property do ALL rigid motions (translations, rotations, reflections) share?
Every rigid motion is an isometry: it preserves distances between points and all angle measures, so the image is always congruent to the original.
2. Point (4, −1) is reflected across the line y = x. What is the image?
Reflecting across y = x swaps the coordinates: (x, y) → (y, x), so (4, −1) → (−1, 4).
3. Point (3, 5) is rotated 180° about the origin. What is the image?
A 180° rotation about the origin sends (x, y) → (−x, −y), so (3, 5) → (−3, −5).
4. Two figures are congruent. What must be true?
The motion-based definition of congruence: two figures are congruent exactly when a sequence of translations, rotations, and/or reflections maps one onto the other.
5. A student rotates an isosceles triangle 180° about its centre and thinks, 'That looks just like a reflection.' What is the best explanation?
Because the triangle is symmetric, a 180° rotation can look similar to a reflected image. But a rotation preserves orientation while a reflection reverses it — the difference only shows up clearly on an asymmetric figure.
✅ Key takeaways
- Translations, rotations, and reflections are the three rigid motions: they move a figure without stretching or distorting it.
- Every rigid motion preserves length and angle measure, so the image is always congruent to the original figure.
- Two figures are congruent exactly when a sequence of rigid motions carries one exactly onto the other.
- Coordinate rules: translation adds (a, b) to every point; reflections flip a sign or swap coordinates; rotations about the origin follow fixed 90°/180°/270° rules.
- A reflection reverses orientation (a flip); a rotation does not — use an asymmetric shape to tell them apart when a figure's symmetry makes them look alike.