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What is the Dilation about the Origin?
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 look and position relative to the origin (the point (0,0)). It's like zooming in or out on a picture on your phone. Every point of the shape moves further from or closer to the origin by a fixed amount, called the scale factor.
Simple Example
Quick Example
Imagine you have a small photo of your family. If you zoom in on it, everyone in the photo gets bigger, but their positions relative to each other don't change. Dilation does the same thing to shapes on a graph. If a point is at (2,3) and you dilate it by a scale factor of 2, its new position will be (4,6), making it twice as far from the origin.
Worked Example
Step-by-Step
Let's dilate a point P(3, 4) about the origin with a scale factor of 2.
Step 1: Identify the coordinates of the point. Here, x = 3 and y = 4.
---Step 2: Identify the scale factor. Here, the scale factor (k) = 2.
---Step 3: To find the new x-coordinate (x'), multiply the original x-coordinate by the scale factor: x' = x * k = 3 * 2 = 6.
---Step 4: To find the new y-coordinate (y'), multiply the original y-coordinate by the scale factor: y' = y * k = 4 * 2 = 8.
---Step 5: The new dilated point P' will have coordinates (x', y').
---Answer: The dilated point P' is (6, 8).
Why It Matters
Dilation is super important in fields like computer graphics and engineering. Game developers use it to zoom characters in and out, and architects use it to scale blueprints. Understanding dilation helps in careers like becoming a software engineer, a graphic designer, or even an urban planner.
Common Mistakes
MISTAKE: Adding the scale factor instead of multiplying. For example, dilating (2,3) by scale factor 2 to get (2+2, 3+2) = (4,5). | CORRECTION: Always multiply each coordinate by the scale factor. For (2,3) with scale factor 2, the correct dilation is (2*2, 3*2) = (4,6).
MISTAKE: Only multiplying one coordinate by the scale factor. For example, dilating (3,5) by scale factor 3 to get (9,5). | CORRECTION: Remember to multiply BOTH the x-coordinate and the y-coordinate by the scale factor. The correct dilation for (3,5) with scale factor 3 is (3*3, 5*3) = (9,15).
MISTAKE: Confusing dilation with translation (sliding the shape). | CORRECTION: Dilation changes the size of the shape and its distance from the origin, while translation only moves the shape without changing its size or orientation.
Practice Questions
Try It Yourself
QUESTION: Dilate the point A(1, 5) about the origin with a scale factor of 3. What are the new coordinates? | ANSWER: (3, 15)
QUESTION: A small square has vertices at (1,1), (1,2), (2,2), and (2,1). If it is dilated by a scale factor of 0.5 about the origin, what are the new coordinates of its vertices? | ANSWER: (0.5, 0.5), (0.5, 1), (1, 1), (1, 0.5)
QUESTION: A point P is at (4, -6). After dilation about the origin, its new position is P'(2, -3). What was the scale factor used for this dilation? | ANSWER: 0.5 (or 1/2)
MCQ
Quick Quiz
If a point (x, y) is dilated about the origin by a scale factor 'k', what are its new coordinates?
(x + k, y + k)
(x * k, y * k)
(x - k, y - k)
(x / k, y / k)
The Correct Answer Is:
B
Dilation involves multiplying each coordinate by the scale factor. Option B correctly shows (x * k, y * k) which scales both the x and y coordinates proportionally.
Real World Connection
In the Real World
When you use Google Maps or any navigation app, zooming in and out on the map is a real-world example of dilation. The map (a representation of an area) gets bigger or smaller, but the relative distances and positions of places like your home, school, or the nearest chai shop remain correct. This scaling helps you understand distances better.
Key Vocabulary
Key Terms
ORIGIN: The point (0,0) on a coordinate plane where the x and y axes intersect. | SCALE FACTOR: The number by which the coordinates of a shape are multiplied to make it larger or smaller. | COORDINATES: A pair of numbers (x, y) that show the exact position of a point on a graph. | DILATION: A transformation that changes the size of a figure but not its shape.
What's Next
What to Learn Next
Great job understanding dilation! Next, you can explore other types of geometric transformations like reflection, rotation, and translation. These concepts build on your understanding of how shapes move and change on a plane, which is crucial for advanced geometry and even computer graphics.
