Dilation Example 1
Graph the rectangle ABCD with coordinates A(-2, 1), B(1, 1), C(1, -1), D(-2, -1). Then dilate the image by a scale factor of 2 with the origin as the center of dilation.
First, we start by plotting the points for the rectangle ABCD.
Next, we take all of the coordinates, and we multiply them by 2:
A'(-4, 2), B'(2, 2), C'(2, -2), D'(-4, -2)
Now we graph the new image with the original image.
Is the new figure an enlargement or reduction of the original image?
Since the new figure is larger and our scale factor was greater than 1, the new image is an enlargement.
Dilation Example 2
Graph the triangle ABC with coordinates A(2, 6), B(2, 2), C(6, 2). Then dilate the image by a scale factor of 1/2 with the origin as the center of dilation.
First, we graph our original triangle in the coordinate plane:
Next, we multiply each coordinate by the scale factor of 1/2. Multiplying by 1/2 is the same as dividing each coordinate by 2:
A'(1, 3), B'(1, 1), C'(3, 1)
Then, we graph the new image with the original image:
Is the new image an enlargement or reduction of our original figure?
Since the scale factor was a fraction less than 1 and the new triangle is smaller than the original, it is a reduction.
Dilation Example 3
Given the original rectangle ABCD and the dilated image A'B'C'D', determine the scale factor and whether an enlargement or reduction occurred.
First, let's list the coordinates of the original figure and the coordinates of the new figure:
A(0, 2), B(3, 2), C(3, 0), D(0, 0)
A'(0, 4), B'(6, 4), C'(6, 0), D'(0,0)
The dilated image is larger than the original image, so an enlargement has taken place.
Now, what has the first set of coordinates been multiplied by to get the dilated image? Since the coordinates have all been multiplied by 2, the scale factor is 2.
Lesson Summary
All right, let's take a moment to review what we've learned. A dilation is an enlargement or reduction of an image by a scale factor. We also learned that the center of dilation is a point about which we are dilating an object. The scale factor is the number that each coordinate is multiplied by to get the new image. An enlargement is when the scale factor is greater than 1, and the new image is larger than the original. A reduction is when the scale factor is less than 1, and the new image is smaller than the original.
To graph an image and find the dilated image we follow the following steps:
- Graph the original figure.
- Multiply the original coordinates by the scale factor.
- Graph the dilated image by plotting the new points.
