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A translation slides every point by the same vector. It preserves distances, angles, parallelism, orientation, and shape, so it is a rigid motion.
Translations model displacement and symmetry and make vector addition concrete in geometry, graphics, and motion.
Intuition and core definition
A translation slides every point by the same vector. It preserves distances, angles, parallelism, orientation, and shape, so it is a rigid motion. No nonzero translation has a fixed point in the plane.
Notation, language, and conditions
Vector $\langle a,b\rangle$ gives horizontal and vertical displacement. Coordinate rule $(x,y)\mapsto(x+a,y+b)$. The vector from original $P$ to image $P\prime$ must be identical for every point.
Why this idea matters
A translation adds one vector to every point, preserving size, orientation, and parallel direction without creating a fixed centre.
A dependable method
- Read the horizontal and vertical components with signs.
- Add $a$ to each $x$-coordinate and $b$ to each $y$-coordinate.
- Keep vertex labels in corresponding order.
- Draw parallel equal-length arrows from originals to images.
- Subtract coordinates to verify every displacement vector matches.
Worked example
Representations and interpretation
A vector arrow displays magnitude and direction; coordinate addition encodes it numerically. Parallel displacement arrows form a family of congruent segments across the entire figure.
Reasoning about variations
A composition of translations adds vectors: translating by $\langle a,b\rangle$ then $\langle c,d\rangle$ equals one translation $\langle a+c,b+d\rangle$. Opposite vectors compose to identity.
Common mistakes
How to check your work
- Compute image minus original for several vertices.
- Compare corresponding side vectors.
- Translate back by the opposite vector.
Practice
- Translate $(-4,7)$ by $\langle6,-2\rangle$.
- What inverse undoes $\langle3,-8\rangle$?
- Does translation change slope?
Answers and brief solutions
Show answers
- $(2,5)$ $-4+6=2$ and $7-2=5$.
- $\langle-3,8\rangle$ Additive opposite components return each coordinate.
- No Parallelism and direction are preserved.
Synthesis and transfer
Moving a coordinate design by a displacement vector should change every vertex by identical horizontal and vertical amounts; subtracting image and original coordinates verifies the rule.
Translating by vector $\langle a,b\rangle$ sends $(x,y)$ to $(x+a,y+b)$, so the difference image minus original is identical for every point. Segment slopes, lengths, and orientation remain unchanged because both endpoints receive the same addition. Composing translations adds their vectors, and translation by the opposite vector is the inverse. A nonzero translation has no fixed point: solving $(x+a,y+b)=(x,y)$ would require $a=b=0$. These properties distinguish a genuine translation from a rotation that may appear to shift one isolated point by the same amount.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Translate $(-4,7)$ by $\langle6,-2\rangle$.
- $-4+6=2$ and $7-2=5$.
End of lesson
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