What is Dilation?

S3-SA2-0197

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Dilation is a transformation that changes the size of a figure but not its shape. It makes a figure larger (enlargement) or smaller (reduction) by a certain scale factor from a fixed point called the center of dilation.

Simple Example

Quick Example

Imagine you have a small photo of your favourite cricketer. If you zoom in on your mobile phone screen, the photo gets bigger, but the cricketer still looks the same, just larger. This 'zooming in' is like dilation, where the photo's size changes but its shape stays the same.

Worked Example

Step-by-Step

Let's dilate a point A(2, 3) with the center of dilation at the origin O(0,0) and a scale factor of 2.
1. Identify the original point: A(2, 3).
---2. Identify the center of dilation: O(0, 0).
---3. Identify the scale factor: k = 2.
---4. To find the new coordinates (A'), multiply each coordinate of the original point by the scale factor.
---5. New x-coordinate = original x-coordinate * scale factor = 2 * 2 = 4.
---6. New y-coordinate = original y-coordinate * scale factor = 3 * 2 = 6.
---7. The dilated point A' is (4, 6).
Answer: The dilated point A' is (4, 6).

Why It Matters

Dilation is super important in many fields! In computer science, it helps scale images on your phone or computer screen. In engineering, it's used to create scaled models of buildings or cars. Knowing dilation can even help you understand how maps are scaled or how special effects in movies make things look bigger or smaller.

Common Mistakes

MISTAKE: Adding the scale factor to the coordinates instead of multiplying. For example, dilating (2,3) by scale factor 2 to (2+2, 3+2) = (4,5). | CORRECTION: Always multiply the coordinates by the scale factor, not add. (2*2, 3*2) = (4,6).

MISTAKE: Forgetting to consider the center of dilation, especially if it's not the origin. | CORRECTION: If the center of dilation is not the origin, first subtract the center's coordinates from the point, then multiply by the scale factor, and finally add the center's coordinates back.

MISTAKE: Confusing enlargement with reduction. | CORRECTION: If the scale factor is greater than 1, it's an enlargement (gets bigger). If the scale factor is between 0 and 1 (a fraction), it's a reduction (gets smaller).

Practice Questions

Try It Yourself

QUESTION: A square has vertices at (1,1), (1,3), (3,3), and (3,1). If it's dilated by a scale factor of 3 with the origin (0,0) as the center, what are the new coordinates of the vertex (1,1)? | ANSWER: (3,3)

QUESTION: A point P(4, -2) is dilated with the origin as the center of dilation and a scale factor of 0.5. What are the coordinates of the dilated point P'? | ANSWER: (2, -1)

QUESTION: A triangle has vertices A(2, 4), B(6, 4), and C(4, 8). If it is dilated with the origin as the center and a scale factor of 1/2, what are the new coordinates of all three vertices? | ANSWER: A'(1, 2), B'(3, 2), C'(2, 4)

MCQ

Quick Quiz

Which of these transformations changes the size of a figure but keeps its shape the same?

Rotation

Reflection

Dilation

Translation

The Correct Answer Is:

Dilation is the only transformation among the options that changes the size of a figure. Rotation, reflection, and translation only change the position or orientation, not the size.

Real World Connection

In the Real World

When you use Google Maps or any navigation app on your phone, you often zoom in or zoom out to see more details or a wider area. This zooming action is a perfect example of dilation in action! The map changes size, but the roads and landmarks stay in proportion. Similarly, graphic designers use dilation to resize images for websites or print.

Key Vocabulary

Key Terms

TRANSFORMATION: A change in the position, size, or shape of a figure | SCALE FACTOR: The ratio by which a figure is enlarged or reduced during dilation | CENTER OF DILATION: The fixed point from which all points of a figure are dilated | ENLARGEMENT: Dilation where the figure gets larger (scale factor > 1) | REDUCTION: Dilation where the figure gets smaller (scale factor between 0 and 1)

What's Next

What to Learn Next

Great job learning about dilation! Next, you can explore other types of transformations like rotation, reflection, and translation. Understanding all these transformations will give you a strong foundation in geometry and help you see how shapes move and change in the real world.