In plain English: 8.G.A.4 is the Common Core grade 8 math standard that defines similar figures with transformations: one figure is similar to another if a sequence of rotations, reflections, translations and dilations takes it onto the other. Students describe such a sequence for two similar figures, using the scale factor for the dilation. It is taught in Grade 8 Math.
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 for Mathematics · Domain: Geometry (G) · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software. Also written as 8.G.4 · Official standard
In 8.G.A.2, students showed that two figures are congruent (same size and same shape) when rotations, reflections and translations take one onto the other. This lesson adds the fourth move from 8.G.A.3, the dilation, which resizes a figure from a center point by a scale factor. The standard defines similar figures this way: a figure is similar to another if a sequence (a list of moves done one after another) of rotations, reflections, translations and dilations takes the first figure onto the second. Similar figures have the same shape, and their sizes can differ.
The second half of the standard asks students to describe a sequence for two figures that are already known to be similar. Students match the vertices (corners), find the scale factor from one pair of matching sides, and then decide which rigid motions (rotations, reflections and translations) finish the job. Because every move in the sequence keeps angles and multiplies all lengths by the same number, students also see why figures whose side ratios differ can never be similar, even if they "look alike."
Learning Objectives
By the end of this lesson, students will be able to:
Explain that a figure is similar to another when a sequence of rotations, reflections, translations and dilations takes the first onto the second
Find the scale factor between two similar figures from a pair of matching sides
Describe a sequence of a dilation and rigid motions that takes one similar figure onto another, and check it on every vertex
Explain why matching angles of similar figures are equal and matching sides share one ratio
Decide from side ratios that two figures are not similar, and explain why no sequence can work
Prior Knowledge Required
Students should already be comfortable with:
Coordinate rules for translations, reflections, rotations and dilations 8.G.A.3
Describing a sequence of rigid motions (rotations, reflections and translations, which keep lengths and angles) between two congruent figures 8.G.A.2
Scale factors from scale drawings: every length is multiplied by the same number 7.G.A.1
Equivalent ratios, such as 6 : 4 and 9 : 6 6.RP.A.3
On grid paper, students draw three rectangles with a corner at the origin: 3 units by 4 units, 6 units by 8 units, and 6 units by 7 units.
Warm-Up Prompt
"Which of the two larger rectangles looks like an enlarged copy of the 3 by 4 rectangle? Draw a diagonal in each rectangle from the origin. What do you notice? Can you describe one move that turns the small rectangle into the copy?"
The 6 by 8 rectangle is a copy: both sides doubled, and its diagonal lies on top of the small rectangle's diagonal. A dilation by 2 from the origin, (x, y) → (2x, 2y), turns one into the other. The 6 by 7 rectangle has one side doubled and the other multiplied by 7/4, so its diagonal points a different way. Tell students that the copy is similar to the small rectangle, and that this lesson gives the exact meaning of that word.
Direct Instruction20 minutes
Build the idea in steps. Students write each step in their notes next to a small sketch.
Recall the dilation (8.G.A.3): a dilation with center O and scale factor k moves every point along the ray (half-line) from O to k times its distance from O. From the origin, (x, y) → (kx, ky). From any other center, find how far the point is from the center across and up, multiply both distances by k, and count them off from the center again.
Similar: a figure is similar to another if a sequence of rotations, reflections, translations and dilations takes the first onto the second. We write △ABC ~ △PQR, read "triangle ABC is similar to triangle PQR." The order of the letters shows which vertices match: A with P, B with Q, C with R.
What a sequence keeps: rigid motions keep every length and angle, and a dilation keeps every angle and multiplies every length by k. So after the whole sequence, matching angles are equal and every pair of matching sides has the same ratio, the scale factor.
Congruent figures are similar too: a sequence with no dilation, or with a dilation by a scale factor of 1, gives a figure of the same size.
Describing a sequence: (1) match the vertices; (2) find k = image side ÷ matching pre-image side, and check it with a second pair of sides; (3) dilate by k; (4) use rigid motions to finish: a turn if matching sides point in different directions, a reflection if the vertex order changed from counterclockwise to clockwise, a translation to slide it into place; (5) check every vertex.
More than one answer: different sequences can take one figure onto the same image. Any sequence that works on every vertex is correct.
Not similar: if two pairs of matching sides have different ratios, no sequence can work, because every move multiplies all lengths by the same number.
Dilation, then reflection (Diagram 1)
Triangle ABC has vertices A(1, 1), B(4, 1) and C(1, 3). Triangle PQR has vertices P(-2, 2), Q(-8, 2) and R(-2, 6). Describe a sequence that takes ABC onto PQR.
Equation: AB = 3 and PQ = 6, AC = 2 and PR = 4, so k = 2. Dilate by 2 from the origin: (2, 2), (8, 2), (2, 6). ABC goes counterclockwise but PQR goes clockwise, so a reflection is needed: reflect over the y-axis to get P, Q and R. So △ABC ~ △PQR.
Rotation, then dilation (Diagram 2)
Triangle STU has vertices S(6, 2), T(10, 2) and U(10, 8). Triangle S′T′U′ has vertices S′(-1, 3), T′(-1, 5) and U′(-4, 5). Describe a sequence that takes STU onto S′T′U′.
Equation: ST = 4 and S′T′ = 2, TU = 6 and T′U′ = 3, so k = 1/2. ST is horizontal but S′T′ is vertical, so the figure turned 90°. Rotate 90° counterclockwise about the origin, (x, y) → (-y, x): (-2, 6), (-2, 10), (-8, 10). Then dilate by 1/2: (-1, 3), (-1, 5), (-4, 5).
Two figures that are not similar
A rectangle is 2 units by 5 units, with vertices (0, 0), (2, 0), (2, 5), (0, 5). Another is 6 units by 12 units, with vertices (1, 1), (7, 1), (7, 13), (1, 13). Are they similar?
Equation: Short sides: 6 ÷ 2 = 3. Long sides: 12 ÷ 5 = 2.4. The ratios differ, so no single scale factor works and the rectangles are not similar. A rule like (x, y) → (3x, 2y) stretches one direction more than the other, so it is not a dilation.
Two correct sequences
Triangle GHI has vertices G(2, 3), H(4, 3) and I(2, 6). Its image has vertices (-1, -1), (3, -1) and (-1, 5). Find two different sequences.
Equation: GH = 2 and the image side is 4, so k = 2. Sequence 1: dilate by 2 centered at G. G stays put; H is 2 right of G, so its image is 4 right of G, (6, 3); I is 3 up from G, so its image is 6 up from G, (2, 9). Then translate (x - 3, y - 4). Sequence 2: dilate by 2 from the origin, giving (4, 6), (8, 6), (4, 12), then translate (x - 5, y - 7). Both land on the same three points.
Use Diagram 1 to show the dashed middle step: after the dilation, the triangle is the right size but faces the wrong way. Use Diagram 2 to show how the direction of one side reveals the turn. Stress that the scale factor goes from the first figure to the second: image length ÷ pre-image length.
Guided Practice15 minutes
Pairs work each problem on grid paper. One partner finds the scale factor and the other plans the rigid motions; together they check every vertex.
Guided practice problems with answers
Problem
Answer
Show that the triangle with vertices (1, 2), (3, 2), (1, 5) is similar to the triangle with vertices (3, 6), (9, 6), (3, 15).
k = 3; a dilation by 3 from the origin is enough
Describe a sequence from the triangle (2, 2), (6, 2), (2, 4) to the triangle (-1, -1), (-3, -1), (-1, -2).
Dilate by 1/2 from the origin: (1, 1), (3, 1), (1, 2). Then rotate 180° about the origin
Is a 3 by 5 rectangle similar to a 9 by 12 rectangle?
No: 9 ÷ 3 = 3 but 12 ÷ 5 = 2.4
Describe a sequence from the triangle (0, 0), (4, 0), (0, 2) to the triangle (1, 3), (9, 3), (1, 7).
Dilate by 2 from the origin: (0, 0), (8, 0), (0, 4). Then translate (x + 1, y + 3)
Listen for students who divide the pre-image side by the image side and get the reciprocal (the flipped fraction, such as 1/2 instead of 2), and for students who stop after the dilation without checking the position. Ask: "Where is your triangle now, and where does it need to be?"
Independent Practice15 minutes
Students work alone, then compare with a partner. Remind them that a different correct sequence is still correct.
Independent practice problems with answers
Problem
Answer
Describe a sequence from the triangle (3, 1), (6, 1), (3, 3) to the triangle (6, -2), (12, -2), (6, -6).
Dilate by 2 from the origin, then reflect over the x-axis
A right triangle has legs 3 and 4. Another has legs 6 and 7. Are they similar?
No: 6 ÷ 3 = 2 but 7 ÷ 4 = 1.75
A square has vertices (0, 0), (4, 0), (4, 4), (0, 4). Describe a sequence to the square (2, 2), (12, 2), (12, 12), (2, 12).
Dilate by 5/2 from the origin, then translate (x + 2, y + 2)
The triangle (4, 4), (8, 4), (4, 10) is dilated by 1/2 from the origin and then translated 5 units left. Find the image and explain why it is similar.
(-3, 2), (-1, 2), (-3, 5). A dilation and a translation were used, so the angles match and each side is half as long
A triangle is reflected over the x-axis and then dilated by 4 from the origin. Is the image similar to the original, and what is the scale factor?
Yes: a reflection and a dilation were used, so it is similar, with scale factor 4
Closure5-10 minutes
Exit ticket: (1) A triangle has sides 5, 12 and 13. A similar triangle has shortest side 15. Find its other two sides. (36 and 39, since k = 3.) (2) Describe a sequence from the triangle (1, 1), (2, 1), (1, 3) to the triangle (-3, 3), (-6, 3), (-3, 9). (Dilate by 3 from the origin, then reflect over the y-axis.) (3) True or false: a rotation followed by a dilation by 1/3 always gives a similar figure. (True.)
Differentiation Strategies
For Struggling Students
Give a step frame: "Scale factor: ____ ÷ ____ = ____. Step 1: dilate by ____. Step 2: ____."
Start with pairs where only a dilation is needed, then add one rigid motion at a time
Let students cut out the smaller figure, enlarge it on grid paper, and move the cutout to find the rigid motions
For Advanced Students
Ask students to find a sequence that goes back from the second figure to the first, and to explain why its scale factor is the reciprocal
Have students prove that any two circles are similar by describing a translation and a dilation
Extension (beyond this standard): show that two triangles with two pairs of equal angles are similar, which previews 8.G.A.5 and HSG.SRT.A.3
Assessment Guidance
What to Look For
A complete answer names the scale factor and how it was found, lists every move with its details (the center and scale factor of the dilation, the angle and direction of a turn, the line of a reflection, the direction and distance of a slide), and checks the final coordinates of every vertex. For "not similar" answers, look for two side ratios that differ. Watch for students who use the reciprocal of the scale factor, who add instead of multiply, who ignore a change in orientation, and who call figures similar because they "look alike."
02
Classroom Activities
3 Activities
1
Similar or Not? Card Sort
15 minPairs
Each pair gets 8 cards. Each card shows two figures by their vertices. Pairs sort the cards into "similar" and "not similar," writing a sequence for each similar pair and two different side ratios for each pair that is not.
Card 2: Not similar: 3/2 and 5/4 are different ratios
Card 3: Similar: dilate by 2 from the origin, then reflect over the y-axis
Card 4: Similar: dilate by 4/3 from the origin, then translate 2 left and 2 down
Card 5: Not similar: only the horizontal side doubled (ratios 2 and 1)
Card 6: Similar: rotate 90° counterclockwise about the origin, then dilate by 1/2
Card 7: Similar with scale factor 1 (congruent): rotate 90° counterclockwise about the origin, then translate 4 right
Card 8: Not similar: 5/3 and 6/3 are different ratios
Discussion Questions
Exactly three cards are not similar. What did those three have in common?
Card 7 has a scale factor of 1. Is it fair to call those two figures similar? Use the definition to decide.
Card 5 doubles one side but not the other. Why is that move not a dilation?
Modification for Distance Learning
Share the cards in an online graphing tool so students can plot both figures and test each move before they sort.
2
Guess the Similarity Sequence
20 minPairs
Partner A draws a small triangle, secretly applies a two-step sequence from the menu, and gives Partner B only the two triangles. Partner B finds a sequence that works. Then partners switch roles.
Move Menu
Dilations from the origin with scale factor 2, 3 or 1/2
Rotations of 90° or 180° about the origin
Reflections over the x-axis or the y-axis
Translations of up to 5 units in each direction
Sample Round
Partner A draws M(1, 1), N(4, 1), L(1, 2), dilates by 2 and rotates 180°, and shows the image (-2, -2), (-8, -2), (-2, -4). Partner B finds k = 6 ÷ 3 = 2 from MN and M′N′, dilates, and sees that the result must be turned 180°. Partner B may also rotate first and dilate second: both orders give the same image here.
Discussion Questions
Did your partner find the same sequence you used? If not, did theirs still work on every vertex?
With both moves centered at the origin, a rotation and a dilation can be done in either order. Try a translation of 1 unit right and a dilation by 2 in both orders. Do you get the same image?
3
Photo Resize Check
15 minGroups of 3
A photo app shows a 4 by 3 photo as the rectangle (0, 0), (4, 0), (4, 3), (0, 3). Each group tests five edit rules and decides which ones keep the photo similar and which ones distort it.
The 5 Edits
Edit 1: (x, y) → (2x, 2y)
Edit 2: (x, y) → (2x, y)
Edit 3: (x, y) → (3x + 1, 3y - 2)
Edit 4: (x, y) → (-y/2, x/2)
Edit 5: (x, y) → (x + 4, 2y)
Answer Key
Edit 1: (0, 0), (8, 0), (8, 6), (0, 6); similar, a dilation by 2
Edit 2: (0, 0), (8, 0), (8, 3), (0, 3); distorted, the ratios are 2 and 1
Edit 3: (1, -2), (13, -2), (13, 7), (1, 7); similar, a dilation by 3 and then a translation (x + 1, y - 2)
Edit 4: (0, 0), (0, 2), (-1.5, 2), (-1.5, 0); similar, a 90° counterclockwise rotation and then a dilation by 1/2
Edit 5: (4, 0), (8, 0), (8, 6), (4, 6); distorted, the ratios are 1 and 2
Discussion Questions
Only Edits 2 and 5 distort the photo. What do their rules have in common?
How can you tell from a rule alone that it keeps the photo similar?
Challenge Variation
Groups write their own edit rule that turns the photo sideways, halves it and moves it into the third quadrant (where both coordinates are negative), and trade rules with another group to check.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Dilation and a Reflection
Triangle ABC with A(1, 1), B(4, 1) and C(1, 3), drawn to scale. Step 1 dilates it by 2 from the origin (dashed triangle, (2, 2), (8, 2), (2, 6)). Step 2 reflects that triangle over the y-axis onto P(-2, 2), Q(-8, 2) and R(-2, 6). So △ABC ~ △PQR with scale factor 2.
Diagram 2: Finding the Turn and the Scale Factor
Triangle STU with S(6, 2), T(10, 2) and U(10, 8), and triangle S′T′U′ with S′(-1, 3), T′(-1, 5) and U′(-4, 5), drawn to scale. Matching sides give the scale factor 1/2, and the horizontal side ST matches the vertical side S′T′, so a 90° counterclockwise rotation about the origin followed by a dilation by 1/2 takes STU onto S′T′U′.
04
Homework Assignment
~30 min
8.G.A.4 Homework: Similar Figures and Sequences
Directions: Draw both figures on grid paper for every problem. Show how you found each scale factor, list the moves of each sequence in order, and check the final coordinates of every vertex.
Part 1: Similar or Not? (Problems 1-3)
Triangle ABC has vertices A(0, 1), B(4, 1) and C(0, 4). Triangle DEF has vertices D(0, 2), E(8, 2) and F(0, 8). (a) Find DE ÷ AB and DF ÷ AC. (b) Are the triangles similar? (c) If so, describe a sequence that takes ABC onto DEF.
Rectangle 1 has vertices (0, 0), (6, 0), (6, 4), (0, 4). Rectangle 2 has vertices (-3, -2), (6, -2), (6, 4), (-3, 4). Rectangle 3 is 8 units by 6 units. (a) Is Rectangle 2 similar to Rectangle 1? If so, give the scale factor and a sequence. (b) Is Rectangle 3 similar to Rectangle 1? Explain.
Maya says the rule (x, y) → (3x, y + 2) shows that two triangles are similar. Test her rule on the triangle (0, 0), (2, 0), (0, 2). Find the image, compare the side lengths, and explain whether she is right.
Part 2: Describe the Sequence (Problems 4-6)
Triangle JKL has vertices J(1, 2), K(4, 2) and L(1, 4). Triangle J′K′L′ has vertices J′(-2, -4), K′(-8, -4) and L′(-2, -8). Find the scale factor and describe a sequence that takes JKL onto J′K′L′.
Triangle PQR has vertices P(3, 6), Q(9, 6) and R(3, 9). Triangle P′Q′R′ has vertices P′(2, -1), Q′(2, -3) and R′(3, -1). (a) Find the scale factor. (b) PQ is horizontal. In which direction does P′Q′ go, and what does that tell you? (c) Describe a sequence and check every vertex.
Triangle XYZ has vertices X(2, 2), Y(4, 2) and Z(2, 3). Its image has vertices (1, 5), (5, 5) and (1, 7). (a) Find the scale factor. (b) Describe a sequence that uses a dilation centered at the origin. (c) Describe a different sequence that uses a dilation centered at X(2, 2).
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Scale Factor
Correct, found from matching sides and checked with a second pair
Correct but not checked, or the reciprocal
Missing or wrong
Sequence
Every move named with its details, in order, and it works on every vertex
Works for some vertices, or a detail is missing
No sequence, or it does not work
Similar or Not
Decision supported by side ratios
Correct decision with a weak reason
Wrong decision or no reason
Drawings
Both figures drawn to scale and labeled
Drawn, but some labels missing
Missing or incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
According to 8.G.A.4, when is figure B similar to figure A?
Answer: C
This is the definition in the standard. Choice A fails because a dilation changes the area: a scale factor of 2 multiplies the area by 4. Choice B adds instead of multiplying, which changes the shape. Choice D is not enough: a square and a long rectangle both have four sides.
Question 2 of 20 · Multiple Choice
A triangle with sides 4, 6 and 8 is dilated by a scale factor of 2.5 and then rotated 90°. What are the side lengths of the final image?
Answer: A
The dilation multiplies every side by 2.5, and the rotation keeps lengths: 10, 15 and 20. Choice B adds 2.5 instead of multiplying. Choice C forgets the dilation. Choice D divides by 2.5.
Question 3 of 20 · Multiple Choice
Which single move takes the triangle (2, 1), (3, 1), (2, 4) onto the triangle (8, 4), (12, 4), (8, 16)?
Answer: B
Every coordinate is multiplied by 4: (8, 4), (12, 4), (8, 16). Choice A works for (2, 1) only: it sends (3, 1) to (9, 4). Choice C uses the change in the y-coordinate, 4 - 1 = 3, instead of a ratio. Choice D uses pre-image ÷ image, the reciprocal.
Question 4 of 20 · Multiple Choice
Which pair of rectangles is NOT similar?
Answer: D
For 4 by 6 and 6 by 8, the ratios are 6 ÷ 4 = 1.5 and 8 ÷ 6 ≈ 1.33, so no single scale factor works. Choice A has ratio 2 for both sides, choice C has ratio 1.5 for both, and choice B is two squares, which are always similar.
Question 5 of 20 · Multiple Choice
A triangle has sides 8, 10 and 12. A similar triangle has shortest side 6. How long is its longest side?
Answer: B
The scale factor is 6 ÷ 8 = 3/4, so the longest side is 12 × 3/4 = 9. Choice A multiplies by 8 ÷ 6, the reciprocal. Choice C subtracts 2 because 8 - 6 = 2. Choice D uses the middle side: 10 × 3/4 = 7.5.
Question 6 of 20 · Multiple Choice
Which rule does NOT always give a figure similar to the original?
Answer: C
Rule C multiplies only the vertical lengths by 2 and just slides the figure sideways, so a square becomes a rectangle 1 by 2. Choice A is a dilation by 3. Choice B is a rotation of 180° and a dilation by 2. Choice D is a 90° clockwise rotation and a dilation by 1/2.
Question 7 of 20 · Multiple Choice
Which move takes the triangle (2, 4), (6, 4), (2, 6) onto the triangle (1, 2), (3, 2), (1, 3)?
Answer: A
Every coordinate is halved: (1, 2), (3, 2), (1, 3). Choice B would double the triangle instead. Choice C works for (2, 4) only: it sends (6, 4) to (5, 2). Choice D would make every y-coordinate negative.
Question 8 of 20 · Multiple Choice
△ABC ~ △XYZ, and a dilation by 3 is part of the sequence from ABC to XYZ. Angle A measures 40°. What is the measure of angle X?
Answer: D
Rigid motions and dilations keep every angle, so matching angles of similar figures are equal: angle X = 40°. Choice A multiplies the angle by the scale factor, and choice B divides it. Choice C adds 3. Only lengths are multiplied by the scale factor.
Question 9 of 20 · Multiple Choice
Which sequence takes the triangle (1, 2), (3, 2), (1, 5) onto the triangle (-2, -4), (-6, -4), (-2, -10)?
Answer: A
Dilating by 2 gives (2, 4), (6, 4), (2, 10), and rotating 180° changes both signs. Choice B gives (2, -4), with the wrong sign on x. Choice C gives (-2, 4), with the wrong sign on y. Choice D uses the wrong scale factor: the side from (1, 2) to (3, 2) has length 2 and its image has length 4, so k = 2, not 3.
Question 10 of 20 · Multiple Choice
Triangle ABC goes counterclockwise. A similar triangle A′B′C′ is twice as large and goes clockwise. Which two moves could take ABC onto A′B′C′?
Answer: B
The size doubled, so a dilation is needed, and the order of the vertices reversed, so a reflection is needed. Rotations and translations keep the order counterclockwise, so choices A and C cannot reverse it. Choice D cannot change the size.
Question 11 of 20 · Multiple Choice
A 4 by 10 rectangle is similar to a rectangle whose short side is 6. How long is the long side of the second rectangle?
Answer: C
The scale factor is 6 ÷ 4 = 3/2, so the long side is 10 × 3/2 = 15. Choice A adds 2 to 10 because 6 - 4 = 2. Choice B multiplies 10 by 4 ÷ 6, the reciprocal. Choice D adds 4 to 10.
Question 12 of 20 · Multiple Choice
Are two congruent figures also similar?
Answer: D
Congruent figures are matched by rigid motions alone, which is a sequence with a dilation by 1, so they fit the definition of similar. Choice A adds a condition the definition does not have. Choices B and C are too narrow: any rigid motions can be used, for any figure.
Question 13 of 20 · Multiple Choice
A dilation with center (1, 1) and scale factor 3 is applied to the point (2, 3). What is the image?
Answer: B
The point is 1 unit right and 2 units up from the center. Three times that is 3 right and 6 up, so the image is (1 + 3, 1 + 6) = (4, 7). Choice A dilates from the origin instead. Choice C multiplies (2, 3) by 3 and then adds the center. Choice D uses a scale factor of 2.
Question 14 of 20 · Multiple Choice
Which statement is always true for two similar figures?
Answer: A
Every move in the sequence keeps angles, so matching angles are equal. Choices B and C are true only when the scale factor is 1, since a dilation changes lengths and areas. Choice D leaves out rotations, reflections and dilations.
Question 15 of 20 · Short Answer
Triangle ABC has vertices A(1, 3), B(3, 3) and C(1, 4). Triangle PQR has vertices P(4, 12), Q(12, 12) and R(4, 16). Show that the triangles are similar and describe a sequence that takes ABC onto PQR.
AB = 2 and PQ = 8; AC = 1 and PR = 4. Both ratios are 4, so k = 4. A dilation by 4 from the origin gives (4, 12), (12, 12), (4, 16), which are P, Q and R. So △ABC ~ △PQR.
Question 16 of 20 · Short Answer
Triangle DEF has vertices D(4, 2), E(10, 2) and F(4, 6). Triangle D′E′F′ has vertices D′(-2, -1), E′(-5, -1) and F′(-2, -3). Describe a sequence that takes DEF onto D′E′F′.
DE = 6 and D′E′ = 3, so k = 1/2. Every coordinate of the image is -1/2 times the original, so rotate 180° about the origin and dilate by 1/2 (in either order). For example, (4, 2) → (-4, -2) → (-2, -1).
Question 17 of 20 · Short Answer
Is the rectangle with vertices (0, 0), (5, 0), (5, 2), (0, 2) similar to the rectangle with vertices (1, 1), (11, 1), (11, 6), (1, 6)? Explain.
No. The first is 5 by 2 and the second is 10 by 5. The long sides give 10 ÷ 5 = 2, but the short sides give 5 ÷ 2 = 2.5. The ratios differ, so no sequence of rigid motions and a dilation can take one onto the other.
Question 18 of 20 · Short Answer
Sam says: "A triangle with sides 3, 4 and 5 is similar to a triangle with sides 5, 6 and 7, because each side is 2 units longer." Explain his mistake.
Similarity multiplies every length by the same scale factor; it does not add. The ratios 5/3, 6/4 and 7/5 are all different, so the triangles are not similar. A triangle similar to the first with shortest side 5 would have sides 5, 20/3 and 25/3.
Question 19 of 20 · Short Answer
Triangle GHI has vertices G(0, 0), H(2, 0) and I(0, 3). Triangle G′H′I′ has vertices G′(5, 1), H′(9, 1) and I′(5, 7). Describe a sequence that takes GHI onto G′H′I′.
GH = 2 and G′H′ = 4, so k = 2. Dilate by 2 from the origin: (0, 0), (4, 0), (0, 6). Then translate (x + 5, y + 1): (5, 1), (9, 1), (5, 7).
Question 20 of 20 · Short Answer
A figure is rotated 90° clockwise about the origin and then dilated by 3 from the origin. (a) Find the image of the point (2, -1). (b) Describe a sequence that takes the image back onto the original figure.
(a) 90° clockwise, (x, y) → (y, -x), gives (-1, -2); dilating by 3 gives (-3, -6). (b) Undo the steps in reverse order: dilate by 1/3, then rotate 90° counterclockwise. This shows the second figure is also similar to the first, with scale factor 1/3.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.G.A.4 mean?
8.G.A.4 means two figures are similar when a sequence of rotations, reflections, translations and dilations takes one onto the other. Students must also describe such a sequence for two figures they are told are similar, naming the scale factor and each rigid motion.
What is the difference between 8.G.A.2 and 8.G.A.4?
8.G.A.2 is about congruence and uses only rigid motions, so the figures have the same size. 8.G.A.4 adds dilations, so the figures have the same shape but can have different sizes. Congruent figures are the special case with scale factor 1.
How do you find the scale factor between two similar figures?
Divide a side of the image by the matching side of the original. Check with a second pair of matching sides: if the two ratios differ, the figures are not similar. A scale factor greater than 1 means the image is larger, and one between 0 and 1 means it is smaller.
Does the order of the moves matter in 8.G.A.4?
Sometimes it does. A rotation or reflection and a dilation that are both centered at the origin can be swapped, but a translation and a dilation usually cannot. Students should apply the moves in the order they wrote them and check every vertex at the end.
Is there only one correct sequence?
No, many sequences can take one figure onto another. Two students may dilate from different centers or use a rotation where another uses two reflections. Any sequence that sends every vertex to its match is correct.
Why are figures that "look alike" not always similar?
Looking alike is not a test. Two rectangles such as 4 by 6 and 5 by 7 look close, but their side ratios differ, so no dilation can match both sides. Similarity needs one scale factor for every pair of matching sides.
What mistakes do students make with 8.G.A.4?
A common mistake is dividing the original side by the image side, which gives the reciprocal of the scale factor. Others add the same amount to each side, forget a reflection when the vertex order has reversed, or stop after the dilation without moving the figure into place.
How does 8.G.A.4 connect to later math?
It gives the meaning of similar used in 8.G.A.5, where students find that two pairs of equal angles make triangles similar, and in 8.EE.B.6, where similar triangles explain slope. In high school geometry, HSG.SRT.A.2 uses the same definition with more formal proofs.
Do dilations in 8.G.A.4 have to be centered at the origin?
No, any point can be the center. The origin makes the coordinate rule simplest: (x, y) → (kx, ky). With another center, students count how far each point is from the center, multiply that distance by k and count again from the center.
How can parents help with 8.G.A.4 at home?
Use a photo on a phone: zooming with two fingers keeps the picture similar, while stretching it in one direction distorts it. Ask your child to explain the difference using the words scale factor and sequence, and to sketch a small shape and an enlarged copy on graph paper.
07
Related Standards
6 standards
These standards connect to 8.G.A.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.G.A.2Prerequisite
Show congruence with a sequence of rotations, reflections and translations
