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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Similar figures share a shape but not necessarily a size: a photo and its enlargement, a model car and the real car. In grade 8 the idea becomes precise. A figure is similar to another if a sequence of rotations, reflections, translations and dilations maps the first onto the second. Adding dilations to the rigid moves allows the size to change while every angle stays the same and every length scales by the same factor.
Students analyze pairs of figures to decide whether they are similar and, if so, describe a sequence that shows it, for example 'dilate by a scale factor of 2 about the origin, then reflect over the y-axis'. Finding the scale factor means dividing a length in the image by the matching length in the original. If two pairs of corresponding sides give different ratios, the figures cannot be similar. Congruence turns out to be the special case of similarity with a scale factor of 1, a connection worth making explicit with students.
Students sometimes think a rectangle 2 by 5 is similar to one 4 by 7 because 2 was added to each side. Similar figures need the same ratio, so 4 by 10 would be similar.
All rectangles share right angles, but their side ratios differ. A 1 by 1 square and a 1 by 3 rectangle are not similar.
From a small figure to a large one the scale factor is greater than 1. Dividing original by image instead gives the reciprocal.
Students compare the longest side of one figure with the shortest of the other. Corresponding sides must be matched in order first.
Triangle JKL has vertices J(1, 1), K(3, 1) and L(1, 4). Triangle J'K'L' has vertices J'(-3, 3), K'(-9, 3) and L'(-3, 12). Show the triangles are similar and describe a sequence.
Answer: The triangles are similar with scale factor 3: dilate by 3 about the origin, then reflect over the y-axis.
Photo enlargement on a projector or a geometry app is a strong opening: students see a picture grow while keeping its shape, then measure to find that every length grew by the same factor. A sorting task of rectangles and triangles, some similar and some only nearly so, builds the habit of checking ratios rather than trusting appearance.
Expect items asking for a sequence that maps one figure onto a similar figure, or for a missing side using the scale factor. Students should state the center of any dilation and always check at least two pairs of corresponding sides.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) A rotation then a dilation
A dilation can change the size while keeping the shape, so including it gives a similar figure. Sequences of only rigid moves give congruent figures.
Answer: 20 (also accepted: 20 cm)
The scale factor is 12 ÷ 3 = 4. The long side is 5 × 4 = 20 cm.
Answer: C) No, the ratios 6/4 and 8/6 are different
Similar figures need equal ratios. 6/4 = 1.5 but 8/6 ≈ 1.33, so the rectangles are not similar.
Answer: 0.4 (also accepted: 2/5)
Scale factor = image length ÷ original length = 4 ÷ 10 = 0.4. A factor less than 1 means B is smaller.
Answer: A) A dilation with scale factor 1 leaves size unchanged, so a rigid sequence counts as a similarity
Similarity allows dilations, and a dilation by 1 changes nothing. So any congruence is also a similarity.
A full lesson with slides, activities and an exit ticket on similarity through transformations, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.G.A.4 with an answer key, ready in about a minute.
Make a worksheet →Turn similarity through transformations into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Congruent figures are related by rigid motions only, so they are the same size. Similar figures may also need a dilation, so they have the same shape but can differ in size.
Divide a length in the image by the corresponding length in the original. All pairs of corresponding sides should give the same value.