The lesson
This lesson teaches Dilations & Similarity. Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
Dilations resize
A dilation multiplies the distance of each point from the center by the scale factor. Scale factor > 1 enlarges; between 0 and 1 shrinks.
A quick way to test factors: divide the number by each candidate. If the quotient is a whole number with no remainder, you found a factor.
Every whole number has at least two factors: 1 and itself. Prime numbers have exactly those two, which is why primes are the building blocks for bigger numbers.
Similar figures
Similar figures have equal corresponding angles and proportional corresponding sides. A dilation always produces a figure similar to the original.
When you study similar figures, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
A point is at (4, 6). Dilate by a scale factor of 1/2 centered at the origin.
- 1Multiply each coordinate by 1/2.
- 2New point: (2, 3).
Why this matters
Dilations & Similarity shows up constantly in resize figures by a scale factor to make similar shapes. It also connects to what you will see on homework, quizzes, and the next unit in this grade.
Teachers often move fast in class. This page is here so you can pause, re-read, and practice until the idea feels familiar, not just until you have memorized a rule for one day.
Common mistakes to avoid
Rushing to the answer without writing steps. Middle-school math rewards clear work, and you catch errors earlier when steps are visible.
Mixing up similar ideas from the same topic. If two terms feel alike, make a two-column note: what is the same, what is different, and one example of each.
Key ideas from this lesson
- Dilations resize
- Similar figures
Video walkthrough
Practice
For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.
Exercise 1
Try it yourselfDilate (6, 8) by scale factor 1/2 from the origin.
Step-by-step solution
- 1x: 6 × 1/2 = 3.
- 2y: 8 × 1/2 = 4.
- 3Image: (3, 4).
Exercise 2
Try it yourselfDilate (2, 5) by scale factor 3 from the origin.
Step-by-step solution
- 1x: 2 × 3 = 6.
- 2y: 5 × 3 = 15.
- 3Image: (6, 15).
Exercise 3
Try it yourselfA segment has length 12. After a dilation with scale factor 2/3, what is its length?
Step-by-step solution
- 1New length = 12 × (2/3).
- 212 × 2/3 = 8.
- 3The image segment is 8 units long.
Exercise 4
Try it yourselfTwo triangles have matching angles of 40°, 60°, and 80°. Are they necessarily similar?
Step-by-step solution
- 1All three pairs of corresponding angles are equal.
- 2AA (angle-angle) similarity applies.
- 3Yes, the triangles are similar.
Exercise 5
Try it yourselfRectangle A has sides 4 and 6. Rectangle B has sides 10 and 15. Is B a dilation of A?
Step-by-step solution
- 1Ratio 10/4 = 2.5 and 15/6 = 2.5.
- 2Same scale factor for both dimensions.
- 3Yes, B is a dilation of A with scale factor 2.5.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Dilations & Similarity
Question 1 of 4
Dilate (10, 4) by a scale factor of 1/2 from the origin. What is the image?