In the previous lesson, students applied dilations on a circular grid. The circular grid provides two levels of scaffolding:
- The radial lines give rays from the center of the grid which help find the dilated image of points on those rays.
- The circles provide a way to measure the distance of points from the center of dilation.
In this lesson, students apply dilations to points with no grid. In order to perform a dilation, three pieces of information are still needed: a center of dilation, a scale factor, and a point which is dilated. Students practice identifying centers, scale factors, and images of dilation. They also use dilations to make perspective drawings.
Performing dilations without a grid engages students in MP1 as they think about the meaning of dilation in terms of the given information (center, scale factor, point being dilated).
- Create a dilation of a figure given a scale factor and center of dilation.
- Explain (orally) the effect of the scale factor on the size of the image of a polygon and its distance from the center of dilation.
- Identify the center, scale factor, and image of a dilation without a circular grid.
Let’s dilate figures not on grids.
Ensure that rulers, index cards, and colored pencils are available in the geometry toolkits.
- I can apply a dilation to a polygon using a ruler.
center of a dilation
The center of a dilation is a fixed point on a plane. It is the starting point from which we measure distances in a dilation.
In this diagram, point \(P\) is the center of the dilation.
A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point. The fixed point is the center of the dilation. All of the original distances are multiplied by the same scale factor.
For example, triangle \(DEF\) is a dilation of triangle \(ABC\). The center of dilation is \(O\) and the scale factor is 3.
This means that every point of triangle \(DEF\) is 3 times as far from \(O\) as every corresponding point of triangle \(ABC\).
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).
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