In4-6, find the scale factor that was used in the dilation and write a rule to represent the dilation.
The Correct Answer and Explanation is :
The correct answer is: (x,y)→(2x,2y)
To find the scale factor of a dilation and write a rule to represent the dilation, let’s first establish what dilation is. Dilation is a transformation that alters the size of a figure while maintaining its shape. It is defined by a scale factor, which is the ratio of the distance from the center of dilation to a point on the figure compared to the distance from the center to the corresponding point on the image.
Step 1: Identify Original and Image Coordinates
Suppose we have a figure with original coordinates (x, y) and the dilated figure has coordinates (x’, y’). For instance, consider the points A(2, 3) dilated to A'(4, 6).
Step 2: Calculate the Scale Factor
The scale factor ( k ) can be calculated using the formula:
[
k = \frac{\text{Length of a side of the dilated figure}}{\text{Length of the corresponding side of the original figure}}
]
Using the coordinates, for point A(2, 3) and A'(4, 6):
[
k_x = \frac{x’}{x} = \frac{4}{2} = 2
]
[
k_y = \frac{y’}{y} = \frac{6}{3} = 2
]
Thus, the scale factor ( k ) is 2.
Step 3: Write the Dilation Rule
The dilation rule can be expressed as:
[
(x, y) \rightarrow (kx, ky)
]
Given our scale factor ( k = 2 ), the rule for dilation can be written as:
[
(x, y) \rightarrow (2x, 2y)
]
Explanation of the Dilation Process
When applying this rule, every point in the original figure is multiplied by the scale factor. In our example, the original coordinates (2, 3) transform to (4, 6). This process preserves the shape of the figure because the angles remain the same and the sides are proportionally scaled.
In essence, dilation allows us to create a similar figure at a different size, centered around a specific point, known as the center of dilation. The scale factor indicates how much larger or smaller the new figure will be compared to the original. Therefore, understanding the scale factor and the dilation rule is crucial in geometry for transforming shapes accurately while retaining their essential properties.