Triangle Dilation With Scale Factor 1/3: Reduction Explained

When dealing with geometric transformations, dilation is a fundamental concept that involves resizing a shape. This resizing is determined by a scale factor, which dictates whether the shape will be enlarged or reduced. In this article, we will delve into the specifics of triangle dilations, particularly when the scale factor is n = 1/3. We will explore the implications of this scale factor on the size of the triangle and clarify the correct statement regarding the dilation.

Dilations: Enlarging or Reducing Shapes

Dilations are transformations that change the size of a geometric figure without altering its shape. This transformation is centered around a fixed point, known as the center of dilation, and the extent of the resizing is determined by the scale factor (n). The scale factor is a crucial value that dictates whether the dilation will result in an enlargement or a reduction of the original figure.

If the scale factor n is greater than 1 (n > 1), the dilation results in an enlargement, meaning the image (the transformed figure) is larger than the original figure (the pre-image). Conversely, if the scale factor n is between 0 and 1 (0 < n < 1), the dilation results in a reduction, making the image smaller than the pre-image. When n = 1, the dilation produces a congruent figure, essentially leaving the size unchanged. A scale factor of n < 0 involves both resizing and a 180-degree rotation about the center of dilation, often referred to as a reflection through the center.

To understand triangle dilations fully, it's also essential to know how the coordinates of the vertices change during the transformation. If a point (x, y) is dilated by a scale factor n with the center of dilation at the origin (0, 0), the new coordinates of the dilated point become ( nx, ny). For example, if a triangle has vertices at A(2, 4), B(6, 2), and C(4, 6), and it is dilated by a scale factor of 1/2, the new vertices would be A'(1, 2), B'(3, 1), and C'(2, 3). This demonstrates a reduction in size, as each coordinate is halved.

Analyzing the Scale Factor n = 1/3

In our specific scenario, we are given a scale factor of n = 1/3. This value is a fraction between 0 and 1, which, as we discussed earlier, indicates a reduction. This is because each side of the triangle in the image will be 1/3 the length of the corresponding side in the original triangle. To illustrate this further, consider a triangle with side lengths of 9 units, 12 units, and 15 units. If this triangle is dilated by a scale factor of 1/3, the resulting triangle will have side lengths of 3 units, 4 units, and 5 units, clearly demonstrating a reduction in size.

The concept of similarity is also crucial when dealing with dilations. When a figure is dilated, the image is similar to the pre-image. This means that the shapes are the same, but their sizes are different. The corresponding angles remain congruent, and the corresponding sides are proportional. In the case of our triangle dilated by a scale factor of 1/3, the angles of the new triangle will be the same as the angles of the original triangle, but the side lengths will be proportionally smaller.

Consider the implications for area and perimeter as well. Since the side lengths are scaled by a factor of 1/3, the perimeter of the dilated triangle will also be 1/3 the perimeter of the original triangle. However, the area changes by the square of the scale factor. In this case, the area of the dilated triangle will be (1/3)² = 1/9 the area of the original triangle. This highlights the non-linear relationship between the scale factor and the area of the dilated figure.

Determining the Correct Statement

Now, let's evaluate the statements provided in the question. The question posits: A triangle is dilated by a scale factor of n = 1/3. Which statement is true regarding the dilation?

Statement A: It is a reduction because n > 1.

This statement is incorrect because our scale factor n = 1/3 is not greater than 1. As we've established, a scale factor greater than 1 indicates an enlargement, not a reduction.

Statement B: It is a reduction because 0 < n < 1.

This statement is correct. Our scale factor n = 1/3 falls between 0 and 1, which is the defining condition for a reduction. This means the dilated triangle will be smaller than the original triangle. This aligns perfectly with our understanding of dilations and scale factors.

Conclusion

In conclusion, when a triangle is dilated by a scale factor of n = 1/3, the correct statement is that it is a reduction because 0 < n < 1. Understanding the relationship between the scale factor and the resulting dilation (enlargement or reduction) is crucial in geometric transformations. By grasping these fundamental concepts, one can accurately predict and analyze the effects of dilations on various shapes and figures. The key takeaway here is that scale factors between 0 and 1 will always lead to a reduction in size, while those greater than 1 will result in an enlargement. This principle applies not only to triangles but to all geometric figures undergoing dilation.