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Common Core Math8th GradeGeometry

8.G.3: Transformations on Coordinates

Also written as 8.G.A.3 and CCSS.MATH.CONTENT.8.G.A.3 · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.

Print 8.G.3 Graph PaperAll 28 8th Grade Math Standards

The Official Wording

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Common Core State Standards, Grade 8, Geometry · Read it at the source · Checked October 14, 2026

What 8.G.3 Means in Plain English

Eighth graders write each move as a coordinate rule. Sliding 4 right adds 4 to x, reflecting across the x-axis turns (3, 5) into (3, −5), a counterclockwise quarter turn about the origin sends (x, y) to (−y, x), and a dilation by 2 from the origin doubles both.

Students predict an image before drawing it, then plot to check. The dilation rule bridges to similarity in 8.G.4, and in high school these rules become functions that take points as inputs and give points as outputs (HSG-CO.2).

8.G.3 I Can Statements

  • I can write the coordinate rule for a translation or reflection.
  • I can find a figure’s coordinates after a rotation about the origin.
  • I can multiply coordinates by a scale factor to dilate a figure.

8.G.3 Worked Example

One small triangle put through four separate moves, so the rules can be compared side by side.

Problem

Triangle JKL has J(1, 2), K(3, 2), L(3, 5). Find its image after each move on its own: a translation 4 left and 1 up, a reflection across the y-axis, a 180° rotation about the origin, and a dilation with scale factor 2 from the origin.

One triangle, four moves
  1. Translation: subtract 4 from every x and add 1 to every y.(x, y) → (x − 4, y + 1): (−3, 3), (−1, 3), (−1, 6)
  2. Reflection across the y-axis: x changes sign, y stays.(x, y) → (−x, y): (−1, 2), (−3, 2), (−3, 5)
  3. A 180° rotation about the origin changes the sign of both coordinates.(x, y) → (−x, −y): (−1, −2), (−3, −2), (−3, −5)
  4. Dilation by 2 from the origin multiplies both coordinates by 2.(x, y) → (2x, 2y): (2, 4), (6, 4), (6, 10)
  5. Compare side lengths. JK is 2 and KL is 3 in the original and in the first three images. Only the dilation changes them.JK = 2, KL = 3, but after the dilation 4 and 6
AnswerThe translation, reflection and rotation move the triangle without changing its size. The dilation doubles every side, to 4 and 6, and keeps every angle.

Check it: Plot two of the images. J(1, 2) and its reflection (−1, 2) sit the same distance, 1 unit, from the y-axis. And (6, 10) lies on the line from the origin through L(3, 5), twice as far out.

Common Mistakes with 8.G.3

Changing the wrong coordinate in a reflection

J(1, 2) reflected across the x-axis written as (−1, 2)

Try this: Ask which coordinate measures distance from the mirror line. Across the x-axis that is y, so only y changes sign: (1, −2).

Adding the scale factor instead of multiplying

Dilation by 2: L(3, 5) → (5, 7)

Try this: Multiply both coordinates by the scale factor, then check that the image point sits on the line from the origin through the original.

Swapping without the sign in a 90° turn

(3, 2) rotated 90° counterclockwise written as (2, 3)

Try this: Rotate one point with tracing paper to test the rule: (3, 2) lands at (−2, 3), in the second quadrant.

Where 8.G.3 Fits

Comes Before6.NS.6Locate points in all four quadrants and see sign changes as reflections across the axes.7.G.1Redraw a scale drawing at a new scale.
Alongside8.G.1Verify that slides, turns and flips keep lengths and angles.8.G.2Use a sequence of rigid motions to show two figures congruent.
Comes Next8.G.4Use dilations and rigid motions together to show figures similar.
HSG-CO.2Describe transformations as functions that take points to points.
HSG-SRT.1Verify the properties of dilations given by a center and a scale factor.

Practice 8.G.3

Graph Paper GeneratorFree ToolFour-quadrant grids with room for a figure and all four of its images, so each coordinate rule gets checked by plotting.Four quadrantsCoordinate grid
Print Graph Paper

Quick Check

  1. Where does (−2, 5) land after a reflection across the x-axis?
  2. A square with corners (0, 0), (2, 0), (2, 2), (0, 2) is dilated by 3 from the origin. What are the new corners and the side length?
  3. Write the rule for a translation 3 right and 2 down, and apply it to (4, −1).

8.G.3: Other Names

8.G.38.G.A.3CCSS.MATH.CONTENT.8.G.A.3

Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Official wording from the Common Core State Standards PDFs at corestandards.org, last checked October 14, 2026. The explanation, I can statements, example, mistakes and practice on this page are EMStudio’s own, written beside the official wording. They are not part of the standard.