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

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:

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

Answers:

1. \(C'(9, 6)\)
2. \(D'(1.5, 1.75)\)