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:

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 $$0.25$$.

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

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