Key takeaway: A translation is a slide in which each point of a shape travels the same distance in the same direction, leaving its size, shape and orientation untouched.
Grade 88.G.A.1Geometry and Measurement
Definition of Translation
A translation is a slide. Picture pushing a book across a desk without turning it: every corner moves along parallel paths by the same amount. Nothing is flipped, rotated or resized, which makes the image congruent to the original. Of all the rigid motions, a translation is the simplest.
On the coordinate plane the slide is described by two numbers, how far to move horizontally and how far to move vertically. Positive horizontal means right, negative means left, positive vertical means up and negative means down. The same pair is applied to every vertex, and connecting the new vertices gives the image. Each point and its image are joined by a segment of identical length and direction.
Formula and notation
A slide by \(a\) horizontally and \(b\) vertically follows the rule
\[(x, y) \to (x + a, \, y + b)\]
The image of point \(A\) is written \(A^{\prime}\), read “A prime.”
Key facts
- Side lengths, angle measures and orientation do not change.
- Every point moves by the identical horizontal and vertical amounts.
- Segments joining each point to its image run parallel to one another.
- Moving left or down puts negative numbers in a translation rule.
Translation example
Triangle \(ABC\) has vertices \(A(7, 1)\), \(B(10, 1)\) and \(C(8, 4)\). Apply a translation of 6 units left and 3 units up. Give the vertices of the image.
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Left 6 and up 3 gives the rule \((x, y) \to (x – 6, y + 3)\).
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\(A(7, 1) \to A^{\prime}(1, 4)\) and \(B(10, 1) \to B^{\prime}(4, 4)\).
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\(C(8, 4) \to C^{\prime}(2, 7)\).
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Check: \(A^{\prime}B^{\prime} = 4 – 1 = 3\) and \(AB = 10 – 7 = 3\). The size is unchanged.
The image has vertices \(A^{\prime}(1, 4)\), \(B^{\prime}(4, 4)\), \(C^{\prime}(2, 7)\).
Common mistake
Watch out
For the same slide, Sol moves \(B(10, 1)\) to \(B^{\prime}(4, -2)\), subtracting 3 from the y-coordinate.
Why it is wrong
“Up 3” means add 3 to y, not subtract it. The correct image is \(B^{\prime}(4, 4)\). Match each direction word to a sign before applying the rule: left is negative x, right positive x, down negative y, up positive y.
Related terms and lessons
- Grade 8 · Translations, Reflections and RotationsLessonWorksheetQuiz
- Geometry · Rigid TransformationsLessonWorksheetQuiz
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