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Scale Factor Problems: Understanding Dilations in Geometry

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Scale Factor Problems: Understanding Dilations in Geometry

In the world of geometry, transformations play a crucial role in understanding how shapes move and change. One such transformation is dilation, which involves resizing a shape without altering its overall form. This article will delve into the concept of dilations, focusing on scale factor problems, a fundamental aspect of geometric transformations.

What are Dilations?

Imagine taking a photograph and zooming in or out. This action is analogous to dilation in geometry. A dilation is a transformation that changes the size of a figure but preserves its shape. The key element in dilation is the **scale factor**, which determines how much the figure is enlarged or reduced.

The Scale Factor

The scale factor, denoted by ‘k’, is a number that indicates the ratio of the corresponding sides of the original figure (pre-image) and the dilated figure (image).

  • **If k > 1:** The dilation is an enlargement, making the image larger than the pre-image.
  • **If 0 < k < 1:** The dilation is a reduction, making the image smaller than the pre-image.
  • **If k = 1:** The dilation is a congruence transformation, meaning the image and pre-image are identical in size and shape.

Center of Dilation

The center of dilation is a fixed point from which all points of the pre-image are dilated. The center of dilation can be any point, including the origin.

Finding the Scale Factor

To find the scale factor, you can use the following formula:

Scale Factor (k) = Length of a side of the image / Length of the corresponding side of the pre-image

Example: Dilating a Triangle

Let’s consider a triangle ABC with vertices A(1, 2), B(3, 1), and C(2, 4). We want to dilate this triangle by a scale factor of 2 about the origin (0, 0).

To find the coordinates of the dilated triangle A’B’C’, we multiply the coordinates of each vertex by the scale factor:

  • A'(2, 4)
  • B'(6, 2)
  • C'(4, 8)

The image triangle A’B’C’ is now twice the size of the original triangle ABC, but it retains the same shape.

Dilating About a Point Other Than the Origin

When dilating about a point other than the origin, we need to follow a slightly different procedure. Here’s how:

  1. **Translate the figure:** Shift the figure so that the center of dilation coincides with the origin.
  2. **Dilate the figure:** Perform the dilation using the scale factor as described earlier.
  3. **Translate back:** Shift the dilated figure back to its original position.

Practice Problems

1. A rectangle with vertices (1, 1), (4, 1), (4, 3), and (1, 3) is dilated by a scale factor of 3 about the origin. Find the coordinates of the dilated rectangle.

2. A triangle with vertices (2, 2), (4, 1), and (3, 4) is dilated by a scale factor of 1/2 about the point (1, 1). Find the coordinates of the dilated triangle.

Conclusion

Dilations are a fundamental geometric transformation that helps us understand how shapes change in size without altering their form. Scale factor problems are essential in understanding the mechanics of dilations and their impact on the dimensions of geometric figures. By mastering these concepts, you’ll gain a deeper understanding of geometric transformations and their applications in various fields, including art, architecture, and engineering.

Mathematics