In this lesson, students apply dilations to polygons on a grid, both with and without coordinates. The grid offers a way of measuring distances between points, especially points that lie at the intersection of grid lines. If point \(Q\) is three grid squares to the right and two grid squares up from \(P\) then the dilation with center \(P\) of \(Q\) with scale factor 4 can be found by counting grid squares: it will be twelve grid squares to the right of \(P\) and eight grid squares up from \(P\). The coordinate grid gives a more concise way to describe this dilation. If the center \(P\) is \((0,0)\) then \(Q\) has coordinates \((3,2)\). The image of \(Q\) after this dilation is \((12,8)\).
Students continue to find dilations of polygons, providing additional evidence that dilations map line segments to line segments and hence polygons to polygons. The scale factor of the dilation determines the factor by which the length of those segments increases or decreases. Using coordinates to describe points in the plane helps students develop language for precisely communicating figures in the plane and their images under dilations (MP6). Strategically using coordinates to perform and describe dilations is also a good example of MP7.
- Create a dilation of a polygon on a square grid given a scale factor and center of dilation.
- Identify the image of a figure on a coordinate grid given a scale factor and center of dilation.
Let’s dilate figures on a square grid.
Print and cut one copy of the blackline master for each student.
- I can apply dilations to figures on a square grid.
- If I know the angle measures and side lengths of a polygon, I know the angles measures and side lengths of the polygon if I apply a dilation with a certain scale factor.
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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