# How to Understand Dilations: A Step-by-Step Guide

Dilation, in its simplest form, is the act of enlarging or reducing a figure while maintaining its original shape. It's like zooming in or out on a camera: the object's proportions remain unchanged, but its overall size varies. In the realm of geometry, dilations are governed by a scale factor. If the scale factor is greater than \(1\), the dilation results in an enlargement. If it's between \(0\) and \(1\), the dilation leads to a reduction. In this exploration, we will unearth the principles behind dilations and understand their applications in geometry.

## Step-by-step Guide: Dilations

**1.** **Identifying the Center of Dilation**:

The point about which the figure dilates. If it’s the origin, the dilation is easier to visualize, but any point can serve as the center.

**2.** **Determining the Scale Factor**:

Denoted by \(k\), the scale factor indicates the magnitude of dilation. If \(k > 1\), the figure enlarges; if \(0 < k < 1\), it shrinks.

**3. Applying Dilation**:

To dilate a point \(P(x, y)\) with respect to the origin and scale factor \(k\), the new coordinates \(P'(x’, y’)\) will be:

\( x’ = kx \)

\( y’ = ky \)

### Examples

**Example 1:**

Dilate the point \(A(2, 3)\) about the origin using a scale factor of \(2\).

**Solution**:

Using the dilation formula:

\( x’ = kx \)

\( y’ = ky \)

For point \(A\):

\( x’ = 2(2) = 4 \)

\( y’ = 2(3) = 6 \)

The new point \(A’\) after dilation is \(A'(4, 6)\).

**Example 2:**

Dilate the point \(B(4, 5)\) about the origin using a scale factor of \(0.5\).

**Solution**:

Using the dilation formula:

\( x’ = kx \)

\( y’ = ky \)

For point \(B\):

\( x’ = 0.5(4) = 2 \)

\( y’ = 0.5(5) = 2.5 \)

The new point \(B’\) after dilation is \(B'(2, 2.5)\).

### Practice Questions:

- Dilate the point \(C(3, 2)\) about the origin using a scale factor of \(3\).
- Dilate the point \(D(6, 7)\) about the origin using a scale factor of \(0.25\).

**Answers**:

- \(C'(9, 6)\)
- \(D'(1.5, 1.75)\)

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