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**preview**shows half of the first page. to view the full**2 pages of the document.**2) Dilations: A dilation enlarges or reduces the size of a figure. Because a dilation does not

preserve size, it is not an isometry. However, dilations do preserve the shapes of figures. The

preimage and the image will have the same shape, but generally not the same size. Two

figures that have the same shape are said to be similar, and since dilations produce images

that are similar to the preimage, dilations are often called similarity transformations.

A dilation can be completely specified by its scale factor and its center. The scale

factor (usually indicated by the letter k) indicates the factor by which the lengths in the preimage

are multiplied to form the image. For an enlargement, k > 1, and for a reduction, 0 < k < 1.

The center of the dilation is the one fixed point in the plane that does not change when

the dilation is applied. Consider the triangle with vertices at (0, 2), (4, 3), and (1, 0). If this

triangle is enlarged by scale factor of 2, the result will depend on what point we choose for the

center of the dilation. If the origin is the center, the image will be the triangle with vertices at (0,

4), (8, 6), and (2, 0). All of the original coordinates are multiplied by 2. If (0, 2) is the center, the

image will be the triangle with vertices at (0, 2), (8, 4), and (2, -2). If (1, 0) is the center, the

image will be the triangle with vertices at (-1, 4), (7, 6), and (1, 0). All three images are the

same size but their positions depend on where the center of the dilation is. Draw these

triangles, along with the preimage, on a piece of graph paper to get a picture of these dilations.

Most of the time, if you have to describe a dilation using coordinate equations or

matrices, the center of the dilation will be the origin, and your task will be quite simple. In

coordinate notation, a dilation with center at the origin and scale factor k is (x’, y’) = (kx, ky). A

dilation with center at the origin and scale factor 3 is written as (x’, y’) = (3x, 3y). The dilation

(x’, y’) = (0.5x, 0.5y) has its center at the origin and a scale factor of 0.5.

Write a coordinate equation for each dilation. (Type in your answers in color.)

1) Dilation with center (0, 0) and scale factor 0.25. (x’, y’) = ( 0.25x, 0.25y)

2) Dilation with center (0, 0) and scale factor 1.5. (x’, y’) = (1.5x, 1.5y)

3) Dilation with center (0, 0) and scale factor 4. (x’, y’) = (4x, 4y)

4) Dilation with center (0, 0) and scale factor 0.6. (x’, y’) = (0.6x,0.6y)

5) Dilation with center (0, 0) and scale factor 8. (x’, y’) = (8x, 8y)

Write a verbal description of each transformation. (Type in your answers in color.)

1) (x’, y’) = (9x, 9y) Dilation with center (0, 0) and scale factor 9.

2) (x’, y’) = (0.4x, 0.4y) Dilation with center (0,0) and scale factor 0.4.

3) (x’, y’) = (5x, 5y) Dilation with center (0,0) and scale factor 5.

4) (x’, y’) = (1.8x, 1.8y) Dilation with center (0,0) and scale factor 1.8.

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