What is a Dilation?

S3-SA2-0454

Grade Level:

Class 6

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

Definition

What is it?

A dilation is a transformation that changes the size of a figure, making it either bigger or smaller, but keeps its shape the same. Think of it like zooming in or zooming out on a photo. It uses a 'scale factor' to decide how much the size changes.

Simple Example

Quick Example

Imagine you have a small photo of your favourite cricket player on your phone. When you pinch your fingers outwards on the screen, the photo becomes bigger. This act of making the photo bigger while keeping its shape is a dilation.

Worked Example

Step-by-Step

Let's say you have a small square drawn on a graph paper with corners at (1,1), (2,1), (2,2), and (1,2). We want to dilate this square by a scale factor of 2, with the center of dilation at the origin (0,0).

1. Identify the coordinates of the original square: A(1,1), B(2,1), C(2,2), D(1,2).
---2. Multiply each coordinate (x,y) by the scale factor (2) to get the new coordinates (x',y').
---3. For point A(1,1): New A' = (1*2, 1*2) = (2,2).
---4. For point B(2,1): New B' = (2*2, 1*2) = (4,2).
---5. For point C(2,2): New C' = (2*2, 2*2) = (4,4).
---6. For point D(1,2): New D' = (1*2, 2*2) = (2,4).
---Answer: The new, dilated square will have corners at (2,2), (4,2), (4,4), and (2,4). It is twice as big as the original square.

Why It Matters

Dilation is super useful in fields like Computer Graphics, where designers use it to resize images and animations for games or movies. Engineers use it to scale down building plans or machine parts. It helps scientists in AI/ML to process images by resizing them for analysis, making it a foundational concept for many cool careers.

Common Mistakes

MISTAKE: Adding the scale factor to coordinates instead of multiplying. | CORRECTION: Always multiply the coordinates by the scale factor for dilation, not add.

MISTAKE: Changing the shape of the figure during dilation. | CORRECTION: Remember, dilation only changes the size (bigger or smaller), but the shape of the figure always stays the same.

MISTAKE: Not considering the center of dilation. | CORRECTION: If the center of dilation is not the origin (0,0), you first subtract its coordinates, then multiply by the scale factor, and finally add the center's coordinates back.

Practice Questions

Try It Yourself

QUESTION: A line segment has endpoints at (2,3) and (4,5). If it is dilated by a scale factor of 3 with the center at the origin (0,0), what are the new coordinates of the endpoints? | ANSWER: (6,9) and (12,15)

QUESTION: A triangle has vertices at P(1,1), Q(3,1), and R(2,4). If it is dilated by a scale factor of 0.5 with the center at the origin, what are the new coordinates of P', Q', and R'? | ANSWER: P'(0.5, 0.5), Q'(1.5, 0.5), R'(1, 2)

QUESTION: A rectangle has corners at (0,0), (5,0), (5,3), and (0,3). It is dilated by a scale factor of 2, but the center of dilation is at (1,1). What are the new coordinates of the corner that was originally at (5,3)? | ANSWER: (9,5)

MCQ

Quick Quiz

Which of the following transformations changes the size of a figure but preserves its shape?

Translation

Rotation

Reflection

Dilation

The Correct Answer Is:

Dilation specifically changes the size of a figure by a scale factor, either making it larger or smaller, while keeping its shape identical. Translation, rotation, and reflection only move or flip the figure without changing its size.

Real World Connection

In the Real World

When you use Google Maps on your phone to zoom in or out on a street map of your city like Bengaluru or Delhi, you are seeing dilation in action. The map gets bigger or smaller, but the roads and landmarks keep their relative positions and shapes, just like a dilation.

Key Vocabulary

Key Terms

SCALE FACTOR: The number by which the size of a figure is multiplied during dilation, making it bigger or smaller. | CENTER OF DILATION: The fixed point from which all points of a figure are stretched or shrunk. | TRANSFORMATION: A change in the position, size, or shape of a figure. | COORDINATES: A set of numbers that shows the exact position of a point on a graph.

What's Next

What to Learn Next

Great job understanding dilation! Next, you can explore other types of geometric transformations like translation, rotation, and reflection. These concepts will help you understand how figures move and change in geometry, opening doors to more advanced topics!