Translations on the Coordinate Plane
Translations on the Coordinate Plane is a Grade 8 math skill from Reveal Math, Course 3, Module 9: Congruence and Similarity. A translation slides every point of a figure the same distance in the same direction, mapping each point (x, y) to a new point (x + a, y + b), where a is the horizontal shift and b is the vertical shift. Positive a moves right, negative a moves left; positive b moves up, negative b moves down. Translations are rigid transformations, so the image is always congruent to the original figure. This concept is foundational in 8th grade geometry because it introduces the formal coordinate notation for transformations that students use throughout modules on congruence, similarity, and beyond.
Key Concepts
A translation is a rigid transformation that slides a figure horizontally, vertically, or both, without changing its size, shape, or orientation.
On the coordinate plane, a translation maps every point $(x, y)$ to a new point $(x + a, y + b)$: $$(x, y) \rightarrow (x + a, y + b)$$.
Common Questions
What is a translation on the coordinate plane?
A translation is a rigid transformation that slides every point of a figure the same distance in the same direction without changing its size or shape. On the coordinate plane, it maps each point (x, y) to (x + a, y + b).
How do you translate a point on the coordinate plane?
To translate a point (x, y), add the horizontal shift a to the x-coordinate and the vertical shift b to the y-coordinate. For example, translating P(2, 5) by the rule (x+3, y-4) gives the image point P'(5, 1).
What do positive and negative values mean in a translation rule?
In the rule (x + a, y + b): a positive a shifts right, negative a shifts left; positive b shifts up, negative b shifts down. Always keep the run positive to avoid direction errors with negative slopes.
Are translations rigid transformations?
Yes. Translations are rigid transformations, meaning they preserve the size and shape of the figure. The image after a translation is always congruent to the original preimage.
When do students learn translations in 8th grade math?
In Grade 8 Reveal Math Course 3, translations are taught in Module 9: Congruence and Similarity, as part of a broader study of rigid transformations that also includes reflections and rotations.
How do you translate a triangle on the coordinate plane?
Apply the translation rule to each vertex individually. With rule (x-2, y+3): vertex A(1,2) maps to A'(-1,5), B(3,5) maps to B'(1,8), and C(4,1) maps to C'(2,4). Connect the new vertices to complete the image.