What is Dilation about the Origin?

S3-SA2-0368

Grade Level:

Class 6

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

Definition

What is it?

Dilation about the origin is a way to make a shape bigger or smaller while keeping its original form and position relative to the center point (the origin). It's like using a zoom feature on a photo, where the center of the zoom stays fixed at the origin (0,0) on a graph.

Simple Example

Quick Example

Imagine you have a small photo of your favourite cricket player on your phone. When you pinch-zoom to make it bigger, the photo gets larger, but the player still looks the same, just scaled up. If the center of your screen is the origin, that's like a dilation. The image gets bigger or smaller, but its 'shape' doesn't change, and it grows/shrinks from that central point.

Worked Example

Step-by-Step

Let's dilate a point A(2, 3) about the origin with a scale factor of 2.

Step 1: Identify the coordinates of the point: A(x, y) = A(2, 3).
---Step 2: Identify the scale factor (k). Here, k = 2.
---Step 3: To dilate a point (x, y) about the origin, multiply each coordinate by the scale factor: A'(x', y') = (k*x, k*y).
---Step 4: Calculate the new x-coordinate: x' = 2 * 2 = 4.
---Step 5: Calculate the new y-coordinate: y' = 2 * 3 = 6.
---Step 6: Write down the new coordinates of the dilated point: A'(4, 6).

Answer: The dilated point is A'(4, 6).

Why It Matters

Dilation is super important in fields like computer graphics, where it's used to resize images and 3D models for games and movies. Engineers use it to scale blueprints and designs, and architects use it to create models of buildings. Even app developers use it to make sure apps look good on different phone screens!

Common Mistakes

MISTAKE: Adding the scale factor instead of multiplying. For example, dilating (2,3) by scale factor 2 to (2+2, 3+2) = (4,5). | CORRECTION: Always multiply each coordinate by the scale factor. So, (2*2, 3*2) = (4,6).

MISTAKE: Only multiplying one coordinate by the scale factor. For example, dilating (2,3) by scale factor 2 to (4,3) or (2,6). | CORRECTION: Remember to multiply BOTH the x-coordinate and the y-coordinate by the scale factor.

MISTAKE: Confusing a scale factor less than 1 with making the shape bigger. | CORRECTION: A scale factor greater than 1 makes the shape larger (enlargement). A scale factor between 0 and 1 (like 0.5 or 1/2) makes the shape smaller (reduction).

Practice Questions

Try It Yourself

QUESTION: Dilate the point B(5, 10) about the origin with a scale factor of 3. | ANSWER: B'(15, 30)

QUESTION: A triangle has vertices P(1, 2), Q(3, 1), and R(2, 4). Dilate this triangle about the origin with a scale factor of 0.5. What are the new coordinates? | ANSWER: P'(0.5, 1), Q'(1.5, 0.5), R'(1, 2)

QUESTION: A square has corners at (2,2), (-2,2), (-2,-2), and (2,-2). If you dilate this square about the origin to get a new corner at (4,4), what was the scale factor used? | ANSWER: The scale factor was 2 (because 2 * 2 = 4).

MCQ

Quick Quiz

If a point (x, y) is dilated about the origin with a scale factor of 'k', what are the new coordinates?

(x + k, y + k)

(x * k, y * k)

(x / k, y / k)

(x - k, y - k)

The Correct Answer Is:

To dilate a point about the origin, you multiply both the x and y coordinates by the given scale factor. Adding, subtracting, or dividing would not result in a correct dilation.

Real World Connection

In the Real World

When you use Google Maps or any navigation app on your phone, and you zoom in or out, you are seeing dilation in action! The map gets bigger or smaller, but the roads and landmarks stay in the same relative positions, just scaled differently. This helps you see more detail or a wider area, just like how ISRO scientists scale satellite images.

Key Vocabulary

Key Terms

ORIGIN: The point (0,0) on a coordinate plane, the fixed center for dilation. | SCALE FACTOR: The number by which you multiply the coordinates to make a shape bigger or smaller. | DILATION: A transformation that changes the size of a figure but not its shape. | ENLARGEMENT: A dilation with a scale factor greater than 1, making the shape bigger. | REDUCTION: A dilation with a scale factor between 0 and 1, making the shape smaller.

What's Next

What to Learn Next

Great job understanding dilation! Next, you can explore other types of transformations like translation (sliding a shape), reflection (flipping a shape), and rotation (turning a shape). These concepts will help you understand how graphics and designs are manipulated in the real world.