Understanding Dilations

TL;DR

Dilations are transformations that change the size of a shape without changing its overall form or orientation. You'll use a scale factor to make the shape bigger or smaller, and a center of dilation as the fixed point from which the sizing occurs. Every point in the original shape is moved along a line from the center, proportional to the scale factor.

1. The Mental Model

Imagine you're using a projector. The original image is on the slide, the light source is the center of dilation, and the image on the screen is the dilation. Moving the projector closer or further away changes the "scale factor" of the projected image.

2. The Core Material

A dilation is a transformation that produces an image that is the same shape as the original, but a different size. It's defined by two things:

1. Center of Dilation: This is a fixed point from which all measurements for the dilation are made. Think of it as the anchor point.
2. Scale Factor (k): This number tells you how much the shape will grow or shrink.
   • If k > 1, the image will be larger (an enlargement).
   • If 0 < k < 1, the image will be smaller (a reduction).
   • If k = 1, the image will be the same size (no change).
   • If k < 0, the image will be dilated and also rotated 180 degrees around the center of dilation.

To perform a dilation, you draw a line from the center of dilation through each point of the original shape. The new point (image) will lie on that line, k times the distance from the center of dilation as the original point.

Let's say your center of dilation is the origin (0,0). If an original point is (x, y) and your scale factor is k, the new point (x', y') will be (kx, ky). If the center of dilation is (a, b), then the formula for a dilated point (x', y') from an original point (x, y) is (a + k(x - a), b + k(y - b)).

Here's how to think about the process:

graph TD
    A["Identify Center of Dilation (C)"] --> B["Identify Original Point (P)"]
    B --> C_prime["Calculate distance from C to P"]
    C_prime --> D["Multiply distance by Scale Factor (k)"]
    D --> E["Draw line from C through P"]
    E --> F["New point (P') is on this line, k times the distance from C"]

Understanding the Scale Factor

Photo by Caroline Cagnin on Pexels

The scale factor doesn't just apply to coordinates; it also applies to lengths and areas:

• Lengths: If the original length of a side is L, the new length L' will be k * L.
• Perimeter: If the original perimeter is P, the new perimeter P' will be k * P.
• Area: If the original area is A, the new area A' will be k² * A.

3. Worked Example

Let's dilate a triangle.
Original Triangle Vertices: A(1, 2), B(3, 1), C(2, 4)
Center of Dilation: Origin O(0, 0)
Scale Factor: k = 2

Step 1: Apply the scale factor to each coordinate.
For point A(1, 2):
A' = (2 * 1, 2 * 2) = (2, 4)

For point B(3, 1):
B' = (2 * 3, 2 * 1) = (6, 2)

For point C(2, 4):
C' = (2 * 2, 2 * 4) = (4, 8)

Result: The new triangle A'B'C' has vertices A'(2, 4), B'(6, 2), C'(4, 8). This is an enlargement, as the scale factor was greater than 1.

4. Key Takeaways

• A dilation changes the size of a shape but keeps its orientation and fundamental form.
• The center of dilation is the fixed point from which all points are scaled.
• The scale factor k determines if the shape gets larger (k > 1), smaller (0 < k < 1), or stays the same (k = 1).
• To find a new point (x', y') after dilation with center (0,0) and scale k, you multiply (x, y) by k to get (kx, ky).
• Lengths and perimeters scale by k, while areas scale by k².

Common Mistakes to Avoid:
- Forgetting to apply the scale factor to all coordinates of all points.
- Confusing the center of dilation with one of the shape's vertices.
- Incorrectly calculating the new coordinates when the center of dilation isn't the origin.
- Thinking that a negative scale factor just means a reduction; it also involves a 180-degree rotation.

5. Now Try It

Dilate the square with vertices P(2, 2), Q(4, 2), R(4, 4), S(2, 4). Use the center of dilation D(1, 1) and a scale factor of k = 0.5. What are the new coordinates of P', Q', R', and S'? Draw both the original and dilated squares on a coordinate plane.

Success looks like: You've found the correct coordinates for P', Q', R', and S' using the formula for a non-origin center, and your drawing shows a smaller square that is centered relative to D(1, 1) compared to the original.