How to Calculate A Dilation with A Negative Scale Factor
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Dilation is a transformation in geometry that changes the size of a figure based on a scale factor. When the scale factor is negative, the figure is not only resized but also reflected across the origin. This guide explains how to calculate dilation with a negative scale factor, provides a calculator, and includes examples.
What is Dilation?
Dilation is a geometric transformation that produces an image that is the same shape as the original but is a different size. The size change is determined by the scale factor, which can be any positive real number. When the scale factor is greater than 1, the image is larger than the original. When the scale factor is between 0 and 1, the image is smaller than the original.
Dilation can be represented using coordinates. For any point (x, y) in the original figure, the corresponding point in the dilated figure is (kx, ky), where k is the scale factor.
Negative Scale Factor
A negative scale factor means the figure is not only resized but also reflected across the origin. This means that the image will be flipped both horizontally and vertically compared to the original figure.
For example, if the original point is (2, 3) and the scale factor is -2, the dilated point will be (-4, -6). The negative sign indicates the reflection across the origin.
Formula
The formula for dilation with a scale factor k is:
For any point (x, y) in the original figure, the corresponding point in the dilated figure is (kx, ky).
When k is negative, the figure is reflected across the origin as well as being resized.
How to Calculate
To calculate the dilation of a figure with a negative scale factor:
- Identify the coordinates of the original figure.
- Choose a negative scale factor k.
- Multiply each x-coordinate by k to get the new x-coordinate.
- Multiply each y-coordinate by k to get the new y-coordinate.
- The resulting coordinates will be the dilated figure.
The negative scale factor will cause the figure to be reflected across the origin as well as being resized.
Example
Let's say we have a triangle with vertices at (1, 2), (3, 4), and (5, 6). We want to dilate this triangle with a scale factor of -2.
Applying the dilation formula:
- (1, 2) → (-2, -4)
- (3, 4) → (-6, -8)
- (5, 6) → (-10, -12)
The dilated triangle will have vertices at (-2, -4), (-6, -8), and (-10, -12). Notice that the figure has been both resized and reflected across the origin.
FAQ
- What happens if the scale factor is negative?
- The figure is not only resized but also reflected across the origin. This means the image will be flipped both horizontally and vertically compared to the original figure.
- Can the scale factor be a fraction?
- Yes, the scale factor can be any real number, including fractions. A scale factor between 0 and 1 will shrink the figure, while a scale factor greater than 1 will enlarge it.
- How does dilation affect the size of the figure?
- The size of the figure is determined by the absolute value of the scale factor. A scale factor of 2 will make the figure twice as large, while a scale factor of 0.5 will make it half as large.
- Is dilation a linear transformation?
- Yes, dilation is a linear transformation because it preserves the straightness of lines and the ratios of distances along them.
- Can dilation be applied to three-dimensional figures?
- Yes, dilation can be applied to three-dimensional figures as well. The formula would be (kx, ky, kz) for any point (x, y, z) in the original figure.