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Common Core Math8th GradeGeometry

8.G.4: Similarity by Transformations

Also written as 8.G.A.4 and CCSS.MATH.CONTENT.8.G.A.4 · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.

Print 8.G.4 Graph PaperAll 28 8th Grade Math Standards

The Official Wording

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

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

What 8.G.4 Means in Plain English

Eighth graders show two figures similar by finding moves that carry one onto the other, resizing included. A 2 by 3 rectangle maps onto a 6 by 9 one by a dilation with scale factor 3, then a slide. Angles stay put, and every length triples.

So a 2 by 3 rectangle and a 4 by 5 rectangle are not similar, since 4 ÷ 2 is 2 but 5 ÷ 3 is not. High school defines similarity by these same transformations and uses it to prove triangle facts (HSG-SRT.2).

8.G.4 I Can Statements

  • I can tell whether two figures are similar using transformations.
  • I can find the scale factor between two similar figures.
  • I can describe a sequence, dilation included, that maps one figure onto another.

8.G.4 Worked Example

A small triangle and one three times its size that also faces the other way, so the sequence needs two moves.

Problem

Triangle PQR has P(1, 1), Q(3, 1), R(1, 2). Triangle STU has S(−3, 3), T(−9, 3), U(−3, 6). Describe a sequence of transformations that maps PQR onto STU.

Mapping a small triangle onto a large one
  1. Compare matching sides to find the scale factor. Both pairs must give the same number.ST ÷ PQ = 6 ÷ 2 = 3, SU ÷ PR = 3 ÷ 1 = 3
  2. Dilate PQR by 3 from the origin: multiply every coordinate by 3.(x, y) → (3x, 3y): (3, 3), (9, 3), (3, 6)
  3. The big triangle points left from S, while this one points right, so flip it across the y-axis.(x, y) → (−x, y): (−3, 3), (−9, 3), (−3, 6)
  4. These are exactly S, T and U, in order, so the sequence works.P → S, Q → T, R → U
AnswerDilate PQR by a scale factor of 3 from the origin, then reflect it across the y-axis. PQR is similar to STU.

Check it: Both triangles have a right angle at the first vertex, P and S, as similar figures must. Measure QR and TU with a ruler: TU comes out three times as long.

Common Mistakes with 8.G.4

Growing every side by the same amount

Triangles with sides 3, 4, 5 and 5, 6, 7 called similar because each side grew by 2

Try this: Divide each new side by its old side. Similar figures share one scale factor, but 5 ÷ 3 and 6 ÷ 4 differ.

Stopping after the dilation

“Dilate by 3” given as the whole answer, with the image still in the wrong quadrant

Try this: Compare the dilated image with the target. If it sits in the wrong place or faces the wrong way, add the slide, turn or flip.

Thinking similar means nearly the same size

“They can’t be similar, one is three times as big.”

Try this: Contrast the two words: congruent means same shape and same size, while similar means same shape at any size, with equal angles.

Where 8.G.4 Fits

Comes Before7.G.1Compute actual lengths from a scale drawing and redraw it at a new scale.7.RP.2Recognize a constant ratio between two quantities.
Alongside8.G.3Write the coordinate rule for a dilation from the origin.8.G.5Use two equal angles to show triangles similar.8.EE.6Use similar triangles to explain why a line’s slope never changes.
Comes Next
HSG-SRT.2Decide whether two figures are similar using similarity transformations.
HSG-SRT.3Prove the angle-angle criterion for similar triangles.

Practice 8.G.4

Graph Paper GeneratorFree ToolFour-quadrant grids large enough to dilate a small figure by 3 and still fit the image on the page.Four quadrantsSmall squares
Print Graph Paper

Quick Check

  1. Is a 3 by 5 rectangle similar to a 9 by 15 rectangle?
  2. Triangle ABC is dilated by a scale factor of ½ from the origin. Where does B(8, −6) land?
  3. One right triangle has legs 3 and 4. Another has legs 7.5 and 10. Are they similar?

8.G.4: Other Names

8.G.48.G.A.4CCSS.MATH.CONTENT.8.G.A.4

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.