Calculating the Distance between A and the Origin

Oct 13, 2025 by windailys.com 105 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 explore the concept of scale factor and how to find it when a triangle is dilated with a center of dilation at the origin.

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 that compares the size of the original figure to the size of the dilated figure.

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 calculated by dividing the distance between two corresponding points on the dilated figure by the distance between the same two points on the original figure. In other words, the scale factor is the ratio of the lengths of the corresponding sides of the two figures.

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, we need to find the distance between the vertex A and the origin, and the distance between the vertex A' and the origin. The distance between two points (x1, y1) and (x2, y2) is given by the formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Calculating the Distance between A and the Origin

The distance between the vertex A(-6, 4) and the origin (0, 0) is given by:

d1 = √((-6 - 0)^2 + (4 - 0)^2) d1 = √((-6)^2 + 4^2) d1 = √(36 + 16) d1 = √52

Calculating the Distance between A' and the Origin

The distance between the vertex A'(-24, 16) and the origin (0, 0) is given by:

d2 = √((-24 - 0)^2 + (16 - 0)^2) d2 = √((-24)^2 + 16^2) d2 = √(576 + 256) d2 = √832

Finding the Scale Factor

The scale factor is the ratio of the distance between the vertex A' and the origin to the distance between the vertex A and the origin. It is given by:

scale factor = d2 / d1 scale factor = √832 / √52 scale factor = √(832 / 52) scale factor = √16 scale factor = 4

Conclusion

In this article, we have explored the concept of scale factor and how to find it when a triangle is dilated with a center of dilation at the origin. We have calculated the distance between the vertex A and the origin, and the distance between the vertex A' and the origin. We have then found the scale factor by dividing the distance between the vertex A' and the origin by the distance between the vertex A and the origin. The scale factor is a ratio that compares the size of the original figure to the size of the dilated figure.

Example

Suppose we have a triangle with a vertex at A(2, 3) that has been dilated with a center of dilation at the origin, so that the vertex of its image is A'(6, 9). How do we find the scale factor of this triangle?

Step 1: Calculate the Distance between A and the Origin

The distance between the vertex A(2, 3) and the origin (0, 0) is given by:

d1 = √((2 - 0)^2 + (3 - 0)^2) d1 = √(2^2 + 3^2) d1 = √(4 + 9) d1 = √13

Step 2: Calculate the Distance between A' and the Origin

The distance between the vertex A'(6, 9) and the origin (0, 0) is given by:

d2 = √((6 - 0)^2 + (9 - 0)^2) d2 = √(6^2 + 9^2) d2 = √(36 + 81) d2 = √117

Step 3: Find the Scale Factor

The scale factor is the ratio of the distance between the vertex A' and the origin to the distance between the vertex A and the origin. It is given by:

scale factor = d2 / d1 scale factor = √117 / √13 scale factor = √(117 / 13) scale factor = √9 scale factor = 3

Conclusion

In this example, we have found the scale factor of a triangle with a vertex at A(2, 3) that has been dilated with a center of dilation at the origin, so that the vertex of its image is A'(6, 9). The scale factor is a ratio that compares the size of the original figure to the size of the dilated figure.

Applications of Scale Factor

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

Similarity: The scale factor is used to determine whether two figures are similar or not. If the scale factor is equal to 1, then the two figures are congruent. If the scale factor is not equal to 1, then the two figures are similar.
Dilation: The scale factor is used to determine the size of a dilated figure. If the scale factor is greater than 1, then the dilated figure is larger than the original figure. If the scale factor is less than 1, then the dilated figure is smaller than the original figure.
Geometry: The scale factor is used to determine the size of a figure in geometry. It is used to calculate 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 of a model and to calculate the stresses and strains on the model.

Conclusion

In conclusion, the scale factor is a ratio that compares the size of the original figure to the size of the dilated figure. It is used to determine whether two figures are similar or not, and to calculate the size of a dilated figure. The scale factor has many applications in geometry and other fields, including similarity, dilation, geometry, and engineering.

Introduction

In our previous article, we explored 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 provide additional examples to help you understand the concept better.

Q1: What is the scale factor of a triangle with a vertex at A(2, 3) that has been dilated with a center of dilation at the origin, so that the vertex of its image is A'(6, 9)?

A1: To find the scale factor, we need to calculate the distance between the vertex A and the origin, and the distance between the vertex A' and the origin. The distance between the vertex A(2, 3) and the origin (0, 0) is √13, and the distance between the vertex A'(6, 9) and the origin (0, 0) is √117. The scale factor is then given by:

scale factor = d2 / d1 scale factor = √117 / √13 scale factor = √(117 / 13) scale factor = √9 scale factor = 3

Q2: How do I 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'(-24, 16)?

A2: To find the scale factor, we need to calculate the distance between the vertex A and the origin, and the distance between the vertex A' and the origin. The distance between the vertex A(-6, 4) and the origin (0, 0) is √52, and the distance between the vertex A'(-24, 16) and the origin (0, 0) is √832. The scale factor is then given by:

scale factor = d2 / d1 scale factor = √832 / √52 scale factor = √(832 / 52) scale factor = √16 scale factor = 4

Q3: What is the scale factor of a triangle with a vertex at A(0, 0) that has been dilated with a center of dilation at the origin, so that the vertex of its image is A'(3, 4)?

A3: To find the scale factor, we need to calculate the distance between the vertex A and the origin, and the distance between the vertex A' and the origin. The distance between the vertex A(0, 0) and the origin (0, 0) is 0, and the distance between the vertex A'(3, 4) and the origin (0, 0) is √(3^2 + 4^2) = √25 = 5. The scale factor is then given by:

scale factor = d2 / d1 scale factor = 5 / 0 scale factor = undefined

Q4: How do I find the scale factor of a triangle with a vertex at A(2, 3) that has been dilated with a center of dilation at the origin, so that the vertex of its image is A'(6, 9) and the scale factor is 2?

A4: To find the scale factor, we need to calculate the distance between the vertex A and the origin, and the distance between the vertex A' and the origin. The distance between the vertex A(2, 3) and the origin (0, 0) is √13, and the distance between the vertex A'(6, 9) and the origin (0, 0) is √117. We are given that the scale factor is 2, so we can set up the equation:

scale factor = d2 / d1 2 = √117 / √13 2 = √(117 / 13) 2 = √9 2 = 3

This is a contradiction, so the given scale factor is not correct.

Q5: 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'(-24, 16) and the scale factor is 2?

A5: To find the scale factor, we need to calculate the distance between the vertex A and the origin, and the distance between the vertex A' and the origin. The distance between the vertex A(-6, 4) and the origin (0, 0) is √52, and the distance between the vertex A'(-24, 16) and the origin (0, 0) is √832. We are given that the scale factor is 2, so we can set up the equation:

scale factor = d2 / d1 2 = √832 / √52 2 = √(832 / 52) 2 = √16 2 = 4

This is a contradiction, so the given scale factor is not correct.

Conclusion

In this article, we have answered some frequently asked questions about scale factor and provided additional examples to help you understand the concept better. We have also shown that the scale factor is a ratio that compares the size of the original figure to the size of the dilated figure, and that it is used to determine whether two figures are similar or not.