How to Find the Scale Factor of a Dilated Triangle

Oct 13, 2025 by miodeya.com 70 views

Introduction

In geometry, dilation is a transformation that changes the size of a figure. When a figure is dilated, its size is increased or decreased by a scale factor. The scale factor is a ratio that compares the size of the original figure to the size of the dilated figure. In this article, we will discuss the concept of scale factor and how to find it when a triangle is dilated with a center of dilation at the origin.

What is a Scale Factor?

A scale factor is a ratio that compares the size of the original figure to the size of the dilated figure. It is a measure of how much the figure has been enlarged or reduced. The scale factor is always greater than or equal to 1, and it is usually represented by the letter k.

How to Find the Scale Factor of a Dilated Triangle

To find the scale factor of a dilated triangle, we need to know the coordinates of the original vertex and the coordinates of the dilated vertex. The scale factor can be found by dividing the distance between the dilated vertex and the center of dilation by the distance between the original vertex and the center of dilation.

Step 1: Find the Distance Between the Original Vertex and the Center of Dilation

The original vertex is $A(-6,4)$ and the center of dilation is the origin $(0,0)$ . To find the distance between the original vertex and the center of dilation, we can use the distance formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where $(x_1, y_1)$ is the original vertex and $(x_2, y_2)$ is the center of dilation.

Plugging in the values, we get:

d = \sqrt{(-6 - 0)^2 + (4 - 0)^2}

d = \sqrt{36 + 16}

d = \sqrt{52}

Step 2: Find the Distance Between the Dilated Vertex and the Center of Dilation

The dilated vertex is $A^{\prime}(-24,16)$ and the center of dilation is the origin $(0,0)$ . To find the distance between the dilated vertex and the center of dilation, we can use the distance formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where $(x_1, y_1)$ is the dilated vertex and $(x_2, y_2)$ is the center of dilation.

Plugging in the values, we get:

d = \sqrt{(-24 - 0)^2 + (16 - 0)^2}

d = \sqrt{576 + 256}

d = \sqrt{832}

Step 3: Find the Scale Factor

The scale factor is the ratio of the distance between the dilated vertex and the center of dilation to the distance between the original vertex and the center of dilation. We can find the scale factor by dividing the distance between the dilated vertex and the center of dilation by the distance between the original vertex and the center of dilation.

k = \frac{d_2}{d_1}

where $d_2$ is the distance between the dilated vertex and the center of dilation and $d_1$ is the distance between the original vertex and the center of dilation.

Plugging in the values, we get:

k = \frac{\sqrt{832}}{\sqrt{52}}

k = \frac{\sqrt{832}}{\sqrt{52}} \times \frac{\sqrt{52}}{\sqrt{52}}

k = \frac{\sqrt{832 \times 52}}{\sqrt{52 \times 52}}

k = \frac{\sqrt{43264}}{\sqrt{2704}}

k = \frac{208}{52}

k = 4

Conclusion

In this article, we discussed the concept of scale factor and how to find it when a triangle is dilated with a center of dilation at the origin. We used the distance formula to find the distance between the original vertex and the center of dilation, the distance between the dilated vertex and the center of dilation, and the scale factor. The scale factor is a ratio that compares the size of the original figure to the size of the dilated figure. It is a measure of how much the figure has been enlarged or reduced. The scale factor is always greater than or equal to 1, and it is usually represented by the letter k.

Example

Suppose we have a triangle with a vertex at $A(-6,4)$ that has been dilated with a center of dilation at the origin so that the vertex of its image is $A^{\prime}(-24,16)$ . To find the scale factor, we can use the distance formula to find the distance between the original vertex and the center of dilation, the distance between the dilated vertex and the center of dilation, and the scale factor.

Applications

The concept of scale factor has many applications in geometry and other fields. For example, it is used in architecture to design buildings and bridges, in engineering to design machines and mechanisms, and in computer graphics to create 3D models and animations.

References

[1] "Geometry" by Michael Artin
[2] "Dilation" by Math Open Reference
[3] "Scale Factor" by Khan Academy

Introduction

In our previous article, we discussed the concept of scale factor and how to find it when a triangle is dilated with a center of dilation at the origin. In this article, we will answer some frequently asked questions about scale factor and dilation.

Q: What is the scale factor of a triangle with a vertex at $A(-6,4)$ that has been dilated with a center of dilation at the origin so that the vertex of its image is $A^{\prime}(-24,16)$ ?

A: The scale factor is 4. To find the scale factor, we can use the distance formula to find the distance between the original vertex and the center of dilation, the distance between the dilated vertex and the center of dilation, and the scale factor.

Q: How do I find the scale factor of a dilated triangle?

A: To find the scale factor of a dilated triangle, you need to know the coordinates of the original vertex and the coordinates of the dilated vertex. You can use the distance formula to find the distance between the original vertex and the center of dilation, the distance between the dilated vertex and the center of dilation, and the scale factor.

Q: What is the difference between dilation and scale factor?

A: Dilation is a transformation that changes the size of a figure, while scale factor is a ratio that compares the size of the original figure to the size of the dilated figure. The scale factor is a measure of how much the figure has been enlarged or reduced.

Q: Can the scale factor be negative?

A: No, the scale factor cannot be negative. The scale factor is always greater than or equal to 1.

Q: How do I use the scale factor to find the coordinates of the dilated vertex?

A: To find the coordinates of the dilated vertex, you can use the scale factor and the coordinates of the original vertex. You can multiply the coordinates of the original vertex by the scale factor to get the coordinates of the dilated vertex.

Q: What are some real-world applications of scale factor and dilation?

A: Scale factor and dilation have many real-world applications, including architecture, engineering, and computer graphics. They are used to design buildings and bridges, machines and mechanisms, and 3D models and animations.

Q: Can I use the scale factor to find the area of a dilated triangle?

A: Yes, you can use the scale factor to find the area of a dilated triangle. The area of a dilated triangle is equal to the area of the original triangle multiplied by the square of the scale factor.

Q: How do I use the scale factor to find the perimeter of a dilated triangle?

A: To find the perimeter of a dilated triangle, you can use the scale factor and the perimeter of the original triangle. You can multiply the perimeter of the original triangle by the scale factor to get the perimeter of the dilated triangle.

Q: Can I use the scale factor to find the volume of a dilated 3D object?

A: Yes, you can use the scale factor to find the volume of a dilated 3D object. The volume of a dilated 3D object is equal to the volume of the original object multiplied by the cube of the scale factor.

Q: How do I use the scale factor to find the surface area of a dilated 3D object?

A: To find the surface area of a dilated 3D object, you can use the scale factor and the surface area of the original object. You can multiply the surface area of the original object by the square of the scale factor to get the surface area of the dilated object.

Conclusion

In this article, we answered some frequently asked questions about scale factor and dilation. We discussed how to find the scale factor of a dilated triangle, how to use the scale factor to find the coordinates of the dilated vertex, and how to use the scale factor to find the area, perimeter, volume, and surface area of a dilated object.

Example

Applications

The concept of scale factor and dilation has many applications in geometry and other fields. For example, it is used in architecture to design buildings and bridges, in engineering to design machines and mechanisms, and in computer graphics to create 3D models and animations.

References

[1] "Geometry" by Michael Artin
[2] "Dilation" by Math Open Reference
[3] "Scale Factor" by Khan Academy