which graph shows a dilation?

Dou Vamel yi

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16-01-2025

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Dilation is a transformation that changes the size of a figure, but not its shape. Understanding how to identify a dilation on a graph is crucial in geometry. This article will explore what a dilation is, how to spot it graphically, and differentiate it from other transformations.

Understanding Dilation

A dilation is a transformation that scales a figure by a certain factor, called the scale factor. The scale factor determines how much larger or smaller the figure becomes.

• Scale factor > 1: The figure is enlarged.
• Scale factor = 1: The figure remains unchanged.
• 0 < Scale factor < 1: The figure is reduced (shrunk).
• Scale factor = 0: The figure becomes a single point.
• Scale factor < 0: The figure is enlarged or reduced and reflected across the center of dilation.

The center of dilation is a fixed point around which the dilation occurs. All points are transformed away from or towards this center point.

Identifying a Dilation on a Graph

To identify a dilation on a graph, look for these key characteristics:

• Similar Shapes: The original figure and its dilated image are similar. This means they have the same shape, but different sizes. Corresponding angles remain congruent.
• Proportional Sides: The ratio of corresponding side lengths in the original figure and its image is constant and equal to the scale factor.
• Collinearity: The center of dilation, a point on the original figure, and its corresponding point on the dilated image are collinear (lie on the same line).

Let's look at examples:

Example 1: Dilation with a Scale Factor > 1

[Insert image here showing a triangle and its enlargement. Clearly label the original triangle, the dilated triangle, and the center of dilation. Include measurements showing proportional sides.]

• Alt Text: "Graph showing a triangle dilated with a scale factor greater than 1. The original and dilated triangles are similar with proportional sides and a common center of dilation."

In this example, the larger triangle is a dilation of the smaller triangle. The sides are proportionally larger, and all points are moved away from the center of dilation.

Example 2: Dilation with a Scale Factor < 1

[Insert image here showing a square and its reduction. Clearly label the original square, the dilated square, and the center of dilation. Include measurements showing proportional sides.]

• Alt Text: "Graph showing a square dilated with a scale factor less than 1. The original and dilated squares are similar with proportional sides and a common center of dilation."

Here, the smaller square is a dilation of the larger square. The sides are proportionally smaller, and all points are moved towards the center of dilation.

Example 3: Not a Dilation

[Insert image here showing a figure that has been rotated or translated, but not dilated. Make sure it's clearly NOT a dilation.]

• Alt Text: "Graph showing a figure that has been rotated, not dilated. The shape and size are different, demonstrating that it is not a dilation."

This example shows a rotation; the figure’s size and shape remain unchanged, indicating it is not a dilation.

How to Determine the Scale Factor

The scale factor (k) can be calculated using the formula:

k = (Length of a side in the dilated image) / (Length of the corresponding side in the original figure)

Differentiating Dilation from Other Transformations

It's important to distinguish dilation from other transformations like translations, rotations, and reflections.

• Translation: A slide; the figure maintains its size and shape, moving to a new location.
• Rotation: A turn around a point; size and shape remain unchanged.
• Reflection: A flip across a line; size and shape remain unchanged.

Only dilation changes the size of the figure while preserving its shape.

Conclusion

Identifying a dilation on a graph requires careful observation of similar shapes, proportional sides, and collinearity of points with the center of dilation. Understanding the scale factor and differentiating dilation from other transformations is essential for mastering geometric transformations. By analyzing these characteristics, you can confidently determine which graph represents a dilation.