Finding the Equation of the Image

Oct 13, 2025 by pynkocean.com 118 views

Introduction

In geometry, a 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. In this article, we will discuss how to find the equation of the image of a line after a dilation of a given scale factor centered at a specific point.

Understanding the Problem

The problem states that line ${$ n $}$ is represented by the equation ${$ 3x + 4y = 20 $}$. We are asked to determine and state the equation of line ${$ p $}$, the image of line ${$ n $}$, after a dilation of scale factor ${$ \frac{1}{3}$}$ centered at the origin.

What is a Dilation?

A 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. In this case, we are dilating line ${$ n $}$ by a scale factor of ${$ \frac{1}{3}$}$ centered at the origin.

How to Dilate a Line

To dilate a line, we need to multiply the coordinates of each point on the line by the scale factor. In this case, we are dilating line ${$ n $}$ by a scale factor of ${$ \frac{1}{3}$}$, so we will multiply the coordinates of each point on the line by ${$ \frac{1}{3}$}$.

Finding the Equation of the Image

To find the equation of the image of line ${$ n $}$, we need to find the equation of the line that passes through the dilated points. We can do this by substituting the coordinates of the dilated points into the equation of the original line.

Step 1: Multiply the Coordinates of Each Point on the Line by the Scale Factor

To dilate line ${$ n $}$, we need to multiply the coordinates of each point on the line by the scale factor of ${$ \frac{1}{3}$}$. Let's call the coordinates of a point on the line ${$ (x, y) $}$. Then, the coordinates of the dilated point will be ${$ (\frac{1}{3}x, \frac{1}{3}y) $}$.

Step 2: Substitute the Coordinates of the Dilated Points into the Equation of the Original Line

Now that we have the coordinates of the dilated points, we can substitute them into the equation of the original line to find the equation of the image.

The equation of the original line is ${$ 3x + 4y = 20 $}$. We can substitute the coordinates of the dilated points ${$ (\frac{1}{3}x, \frac{1}{3}y) $}$ into this equation to get:

${$ 3(\frac{1}{3}x) + 4(\frac{1}{3}y) = 20 $}$

Simplifying this equation, we get:

${$ x + \frac{4}{3}y = 20 $}$

Step 3: Write the Equation of the Image in Standard Form

The equation of the image is in the form ${$ Ax + By = C $}$. We can rewrite this equation in standard form by multiplying both sides of the equation by 3 to eliminate the fraction.

${$ 3(x + \frac{4}{3}y) = 3(20) $}$

Simplifying this equation, we get:

${$ 3x + 4y = 60 $}$

Conclusion

In this article, we discussed how to find the equation of the image of a line after a dilation of a given scale factor centered at a specific point. We used the equation of the original line and the scale factor to find the equation of the image. We also showed how to write the equation of the image in standard form.

Final Answer

The equation of the image of line ${$ n $}$ after a dilation of scale factor ${$ \frac{1}{3}$}$ centered at the origin is ${$ 3x + 4y = 60 $}$.

References

[1] "Dilation" by Math Open Reference. Retrieved February 2023.
[2] "Similarity Transformations" by Khan Academy. Retrieved February 2023.
[3] "Equations of Lines" by Math Is Fun. Retrieved February 2023.
Dilation of a Line: Q&A ==========================

Introduction

In our previous article, we discussed how to find the equation of the image of a line after a dilation of a given scale factor centered at a specific point. In this article, we will answer some frequently asked questions about dilation of a line.

Q: What is dilation?

A: 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.

Q: What is the scale factor in dilation?

A: The scale factor is a number that determines how much the figure is enlarged or reduced. In the case of dilation of a line, the scale factor is used to multiply the coordinates of each point on the line.

Q: How do I dilate a line?

A: To dilate a line, you need to multiply the coordinates of each point on the line by the scale factor. This will give you the coordinates of the dilated points.

Q: What is the equation of the image of a line after dilation?

A: The equation of the image of a line after dilation is found by substituting the coordinates of the dilated points into the equation of the original line.

Q: How do I write the equation of the image in standard form?

A: To write the equation of the image in standard form, you need to multiply both sides of the equation by the scale factor.

Q: What is the difference between dilation and translation?

A: Dilation is a transformation that changes the size of a figure, while translation is a transformation that moves a figure from one position to another without changing its size.

Q: Can I dilate a line by a scale factor of 0?

A: No, you cannot dilate a line by a scale factor of 0. A scale factor of 0 would mean that the line is not changed at all, which is not a dilation.

Q: Can I dilate a line by a negative scale factor?

A: Yes, you can dilate a line by a negative scale factor. However, this will result in a reflection of the line across the origin.

Q: How do I dilate a line by a scale factor of 1?

A: Dilating a line by a scale factor of 1 means that the line is not changed at all. This is not a dilation, but rather the original line.

Q: Can I dilate a line by a scale factor of -1?

A: Yes, you can dilate a line by a scale factor of -1. This will result in a reflection of the line across the origin.

Conclusion

In this article, we answered some frequently asked questions about dilation of a line. We hope that this article has helped to clarify any confusion and provide a better understanding of dilation of a line.

Final Answer

The equation of the image of line ${$ n $}$ after a dilation of scale factor ${$ \frac{1}{3}$}$ centered at the origin is ${$ 3x + 4y = 60 $}$.

References

[1] "Dilation" by Math Open Reference. Retrieved February 2023.
[2] "Similarity Transformations" by Khan Academy. Retrieved February 2023.
[3] "Equations of Lines" by Math Is Fun. Retrieved February 2023.