What Is The Scale Factor Of A Triangle With A Vertex At A ( − 6 , 4 A (-6, 4 A ( − 6 , 4 ] That Has Been Dilated With A Center Of Dilation At The Origin So That The Vertex Of Its Image Is A ′ ( − 24 , 16 A^{\prime}(-24, 16 A ′ ( − 24 , 16 ]?

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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 of the lengths of the corresponding sides of the original figure and its image. In this article, we will discuss how to find 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 $A^{\prime}(-24, 16)$.

Understanding Dilation

Dilation is a transformation that changes the size of a figure. It is a type of similarity transformation that preserves the shape of the figure but changes its size. When a figure is dilated, its size is increased or decreased by a scale factor. The scale factor is a ratio of the lengths of the corresponding sides of the original figure and its image.

Finding the Scale Factor

To find 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 $A^{\prime}(-24, 16)$, we need to find the ratio of the lengths of the corresponding sides of the original figure and its image.

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

The distance between the original vertex $A (-6, 4)$ and the center of dilation $O (0, 0)$ is given by:

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

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 Image Vertex and the Center of Dilation

The distance between the image vertex $A^{\prime}(-24, 16)$ and the center of dilation $O (0, 0)$ is given by:

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

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

Plugging in the values, we get:

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

$$d^{\prime} = \sqrt{576 + 256}$$

$$d^{\prime} = \sqrt{832}$$

Step 3: Find the Scale Factor

The scale factor is the ratio of the lengths of the corresponding sides of the original figure and its image. It is given by:

$$k = \frac{d^{\prime}}{d}$$

Plugging in the values, we get:

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

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

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

$$k = \frac{\sqrt{43264}}{52}$$

$$k = \frac{208}{52}$$

$$k = 4$$

Conclusion

In this article, we discussed how to find 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 $A^{\prime}(-24, 16)$. We found that the scale factor is 4, which means that the size of the triangle has been increased by a factor of 4.

Applications of Scale Factor

The scale factor has many applications in geometry and other fields. Some of the applications include:

• Similarity: The scale factor is used to determine the similarity of two figures. If the scale factor is equal to 1, the figures are congruent. If the scale factor is greater than 1, the figures are similar but not congruent.
• Dilation: The scale factor is used to determine the scale of a dilation. If the scale factor is greater than 1, the dilation is an enlargement. If the scale factor is less than 1, the dilation is a reduction.
• Geometry: The scale factor is used to determine the size of a figure. It is used to find the perimeter, area, and volume of a figure.
• Engineering: The scale factor is used in engineering to design and build models of machines and structures. It is used to determine the size and scale of a model.
• Computer Graphics: The scale factor is used in computer graphics to create 3D models and animations. It is used to determine the size and scale of a model.

Final Thoughts

In conclusion, the scale factor is an important concept in geometry and other fields. It is used to determine the similarity of two figures, the scale of a dilation, and the size of a figure. It has many applications in geometry, engineering, and computer graphics.

Q1: What is a scale factor?

A1: A scale factor is a ratio of the lengths of the corresponding sides of two similar figures. It is used to determine the size and scale of a figure.

Q2: How do I find the scale factor of a figure?

A2: To find the scale factor of a figure, you need to find the ratio of the lengths of the corresponding sides of the original figure and its image. You can use the formula:

$$k = \frac{d^{\prime}}{d}$$

where $d$ is the distance between the original vertex and the center of dilation, and $d^{\prime}$ is the distance between the image vertex and the center of dilation.

Q3: What is the difference between a scale factor and a dilation factor?

A3: A scale factor and a dilation factor are the same thing. They are both used to determine the size and scale of a figure.

Q4: Can a scale factor be greater than 1?

A4: Yes, a scale factor can be greater than 1. If the scale factor is greater than 1, the dilation is an enlargement. If the scale factor is less than 1, the dilation is a reduction.

Q5: How do I use a scale factor to determine the size of a figure?

A5: To use a scale factor to determine the size of a figure, you need to multiply the length of the corresponding side of the original figure by the scale factor. For example, if the scale factor is 2 and the length of the corresponding side of the original figure is 5, the length of the corresponding side of the image figure is 10.

Q6: Can a scale factor be negative?

A6: No, a scale factor cannot be negative. A scale factor is always a positive ratio.

Q7: How do I use a scale factor to determine the similarity of two figures?

A7: To use a scale factor to determine the similarity of two figures, you need to compare the scale factors of the two figures. If the scale factors are equal, the figures are congruent. If the scale factors are not equal, the figures are similar but not congruent.

Q8: Can a scale factor be a fraction?

A8: Yes, a scale factor can be a fraction. If the scale factor is a fraction, it means that the dilation is a reduction.

Q9: How do I use a scale factor to determine the perimeter and area of a figure?

A9: To use a scale factor to determine the perimeter and area of a figure, you need to multiply the perimeter and area of the original figure by the scale factor. For example, if the scale factor is 2 and the perimeter of the original figure is 10, the perimeter of the image figure is 20.

Q10: Can a scale factor be used to determine the volume of a figure?

A10: Yes, a scale factor can be used to determine the volume of a figure. To use a scale factor to determine the volume of a figure, you need to multiply the volume of the original figure by the scale factor. For example, if the scale factor is 2 and the volume of the original figure is 10, the volume of the image figure is 20.

Conclusion

In conclusion, a scale factor is an important concept in geometry and other fields. It is used to determine the size and scale of a figure, the similarity of two figures, and the perimeter, area, and volume of a figure. It has many applications in geometry, engineering, and computer graphics.