Add a student-friendly definition in the Binder Page editor.
Similarity and Dilations
Students connect dilations to similarity, use scale factors and proportional relationships, and apply triangle similarity criteria.
Find It. Learn It. Master It.
Beginner
15–20 minutes
Triangles & Transformations
Triangles & Transformations
Page 6 of 8
Inside Cover
Similarity and Dilations
At a Glance
Binder SectionTriangles & Transformations
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
How does a dilation change lengths while preserving angle measures and shape?
Build Understanding
Similar figures have the same shape: corresponding angles are congruent and corresponding side lengths are proportional. A dilation centered at a point multiplies every distance from the center by scale factor k.
If k > 1, the image is an enlargement; if 0 < k < 1, it is a reduction. Perimeter scales by k, while area scales by k squared.
Triangles are similar by AA, SAS similarity, or SSS similarity. Order corresponding vertices carefully, write a proportion using matching sides, and solve for unknown lengths.
A sequence of rigid motions and a dilation demonstrates similarity. Similarity also explains indirect measurement, slope relationships, and right-triangle trigonometry.
Visual Model
4Thousands
3Hundreds
2Tens
7Ones
4,327 = 4,000 + 300 + 20 + 7
Goals
Learning Targets
- Find and interpret a scale factor.
- Use dilations to produce similar figures.
- Match corresponding angles and sides.
- Apply AA, SAS, and SSS similarity.
- Solve proportions for missing lengths.
- Distinguish length, perimeter, and area scale factors.
Click each word
Interactive Vocabulary
Add a student-friendly definition in the Binder Page editor.
Add a student-friendly definition in the Binder Page editor.
Add a student-friendly definition in the Binder Page editor.
Reveal one step at a time
Worked Examples
Example 1
Dilate (6,−3) from the origin by 2/3
1
Multiply coordinates by 2/3.
2
Image is (4,−2).
Example 2
Similar triangles have sides 6 and 9 corresponding to 10 and x
1
6/10=9/x.
2
x=15.
Ask and explain
Common Questions
Are all congruent figures similar? Yes, with scale factor 1.
Does similarity require equal side lengths? No; side lengths are proportional.
What happens to area? It is multiplied by k squared.
Does AA prove congruence? No, but it proves triangle similarity.
Notice and correct
Common Misconceptions
Adding the scale factor instead of multiplying
A dilation multiplies distances from the center.
Using noncorresponding sides in proportions
Mark matching angles first.
Assuming similar means congruent
Congruence requires scale factor 1.
Learn Through Video
Watch, pause, and explain
Use the chapter markers, key vocabulary, and reflection prompts to stay actively engaged while you watch.
Not Started
Build Your Skills
Practice until you can explain it
Move from a quick warm-up to guided practice, independent work, and deeper challenges.
0%
0 of 4 stages complete
Warm-Up
Guided
Independent
Challenge
1
Quick Start
Warm-Up
A side of 8 is dilated by 1.5.
2
We Do
Guided Practice
A side 4 maps to 10. Answer: scale factor 2.5.
Dilate (3,-2) by 1/2 about the origin. Answer: (1.5,-1).
A perimeter of 18 is dilated by k = 3. Answer: 54.
An area of 20 is dilated by k = 2. Answer: 80.
Two triangles have two congruent angle pairs. Answer: similar by AA.
3
You Do
Independent Practice
Are 3-4-5 and 6-8-10 triangles similar?
Find a missing side when scale factor is 3/4.
Explain AA similarity.
4
Stretch Your Thinking
Challenge & Real-World Practice
Use a mirror or shadow method to estimate an inaccessible height. Draw similar triangles, collect plausible measurements, and justify the proportion.
Practice Complete
Continue to My Math Journal →
Nice work!
You completed the Practice It learning path.
IXL
Identify Similar Figures
Open resourceDilations and Scale Factors
Open resourceTriangle Similarity
Open resourceSimilar Triangles and Indirect Measurement
Open resourceKhan Academy
Similarity Unit
Open resourceDilations
Open resourceSolving Similar Triangles
Open resourceDeltaMath
DeltaMath: Dilations
Open resourceDeltaMath: Similar Figures
Open resourceDeltaMath: Triangle Similarity
Open resourceDeltaMath: Indirect Measurement
Open resource
BUILD YOUR PERSONAL MATH LIBRARY
Save today’s lesson and grow your MathBinder
Collect lesson resources, revisit recent pages, build review packets, and watch your binder grow over time.
My Binder
0
Saved Items in My Binder
Start your collection
Today’s Lesson
0 resources collected
Printable Resources
Available Now
Choose a Printable ↓
Interactive Notebook Printables
Choose any notebook page below and select Print Blank Copy to complete it by hand.
- Lesson-specific pages
- Clean print layout
- Add to a physical binder
Your Collection
Recently Added
Your saved items will appear here.Build a binder you can use all year.
Binder Preview
My MathBinder
MathBinder
Find It. Learn It. Master It.
0 saved items
Number SystemRatiosAlgebraGeometry
Study Tools
Build a Review Packet
My Collected Lessons
Create a printable list of the lessons saved in this browser.
Favorite This Lesson
Save this page for quick access from your MathBinder dashboard.
Continue Learning
Move to another published lesson in this Binder Section.
Open Geometric Figures and Relationships →Milestones
Your Binder Achievements
Finish Strong
Go to My Math Journal →
Learned. Watched. Practiced. Saved.
Now capture the most important idea in My Math Journal.
Parent Tip
Spend five minutes reviewing one saved lesson together each week. Ask your child to explain one example aloud.
MY PRIVATE MATH SPACE
My Math Journal
Capture your thinking, questions, and reflections. Everything is saved automatically on this device.
Saved on this device
Ready
Confidence Check
How confident do you feel?
Awesome! You’re ready to teach someone else!
Quick Reflection
Complete the thought
My Math Notes
0 words
What do you want to remember?
Learning Record
0 entries
My Journal History
Revisit reflections saved from other MathBinder lessons on this device.
Your journal history will grow as you reflect.
Complete this lesson’s journal to begin your learning record.
Your Math Journal is automatically saved on this device.
Clearing your browser data will erase your saved journal entries.
Show What You Know
Can you do this on your own?
Complete each question without hints. Your result is saved privately on this device.
5–8
Minutes
0 of 5 answered
0%
Before you begin, I can…
- I identify corresponding parts before forming ratios.
- I apply the scale factor consistently.
- I verify equal angles and proportional side lengths.
1
Question 1
Which statement shows the key idea in this lesson?
2
Question 2
Which habit best supports accuracy?
3
Question 3
What should a student do after solving?
4
Question 4
Which explanation is strongest?
5
Question 5
How can an error be found?
Confidence Check
How confident do you feel?
Mastery Result
0%
Your Results
Recommended Next Step
Review and try again
Badge EarnedSimilarity and Dilations Master
Lesson Complete
Great job on Similarity and Dilations!
You reviewed the lesson, practiced the skill, and completed the mastery check.