What is the scale factor of the dilation shown?
The Correct Answer and Explanation is :
To determine the scale factor of a dilation, we compare the dimensions of the original figure to those of its dilated image. The scale factor (k) is calculated by dividing the length of a side in the dilated image by the corresponding side in the original figure.
Understanding Dilation and Scale Factor
Dilation is a transformation that alters the size of a figure while maintaining its shape. This process involves a center of dilation and a scale factor. The center of dilation is a fixed point in the plane, and the scale factor determines how much the figure is enlarged or reduced.
Calculating the Scale Factor
The scale factor (k) is the ratio of the size of the new image to the size of the original image. Mathematically, it is expressed as:
[ k = \frac{\text{Dimension of the dilated image}}{\text{Dimension of the original figure}} ]
For example, if a side of the original figure measures 4 units and the corresponding side of the dilated image measures 8 units, the scale factor is:
[ k = \frac{8}{4} = 2 ]
This indicates that the image is an enlargement with a scale factor of 2.
Properties of Dilated Figures
Dilated figures retain several properties:
- Shape Preservation: The shape of the figure remains unchanged; only the size is altered.
- Angle Congruence: All corresponding angles between the original and dilated figures are congruent.
- Proportional Sides: The lengths of corresponding sides are proportional to the scale factor.
- Parallelism: Parallel lines in the original figure remain parallel in the dilated image.
- Midpoints: The midpoints of corresponding sides in the original and dilated figures coincide.
Determining the Scale Factor from Coordinates
If the figures are represented on a coordinate plane, the scale factor can be determined by comparing the distances between corresponding points. For instance, if point A has coordinates (x₁, y₁) in the original figure and point A’ has coordinates (x₂, y₂) in the dilated image, the scale factor can be calculated as:
[ k = \frac{\text{Distance from center of dilation to A’}}{\text{Distance from center of dilation to A}} ]
This method is particularly useful when the center of dilation is not at the origin.
Example
Consider a triangle with vertices at A(1, 2), B(3, 4), and C(5, 6). After dilation with a scale factor of 2, the new vertices are A'(2, 4), B'(6, 8), and C'(10, 12). The distances from the center of dilation (assumed to be the origin) to points A, B, and C are:
- Distance to A: √(1² + 2²) = √5
- Distance to B: √(3² + 4²) = 5
- Distance to C: √(5² + 6²) = √61
The distances to points A’, B’, and C’ are:
- Distance to A’: √(2² + 4²) = √20
- Distance to B’: √(6² + 8²) = 10
- Distance to C’: √(10² + 12²) = √244
The ratios of the distances from the center of dilation to the corresponding points in the dilated image and the original figure are:
- A’:A: √20/√5 = √4 = 2
- B’:B: 10/5 = 2
- C’:C: √244/√61 = 2
Since the ratios are consistent, the scale factor is confirmed to be 2.
Conclusion
Understanding the scale factor is essential in geometry, as it allows for the analysis of how figures change under dilation. By calculating the scale factor, one can determine the degree of enlargement or reduction of a figure, ensuring that the transformation maintains the figure’s shape and proportionality.
For a visual explanation, you might find the following video helpful: