Reflection of a Line Segment Across a Line

Oct 13, 2025 by windailys.com 77 views

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Introduction

In geometry, a line segment is a part of a line that is bounded by two distinct points. When a line segment is reflected across a line, its image is formed by the same distance and direction from the line of reflection as the original line segment. In this article, we will explore the concept of reflecting a line segment across a line and determine which reflection will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across a Line

A reflection across a line is a transformation that flips a line segment over a line. The line of reflection is called the axis of reflection. When a line segment is reflected across a line, its image is formed by the same distance and direction from the line of reflection as the original line segment.

Properties of Reflection

The line of reflection is the perpendicular bisector of the line segment.
The image of the line segment is the same distance and direction from the line of reflection as the original line segment.
The line segment and its image are congruent.

Reflection of a Line Segment Across a Line

To reflect a line segment across a line, we need to find the axis of reflection. The axis of reflection is the line that passes through the midpoint of the line segment and is perpendicular to the line segment.

Finding the Axis of Reflection

To find the axis of reflection, we need to find the midpoint of the line segment and the slope of the line segment.

The midpoint of the line segment is the average of the x-coordinates and the average of the y-coordinates of the endpoints.
The slope of the line segment is the change in y-coordinates divided by the change in x-coordinates.

Calculating the Midpoint and Slope

Let's calculate the midpoint and slope of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

The midpoint is ${\left(\frac{-4-6}{2},\frac{-6+4}{2}\right)=(-5,-1)}$ .
The slope is ${\frac{4-(-6)}{-6-(-4)}=\frac{10}{-2}=-5}$ .

Finding the Axis of Reflection

The axis of reflection is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ . The equation of the axis of reflection is ${y-(-1)=-5(x-(-5))}$ or ${y=-5x-26}$ .

Reflection Across the Axis of Reflection

Now that we have found the axis of reflection, we can reflect the line segment across the axis of reflection.

Reflection Across the Axis of Reflection

To reflect the line segment across the axis of reflection, we need to find the image of the line segment. The image of the line segment is the same distance and direction from the axis of reflection as the original line segment.

Calculating the Image of the Line Segment

Let's calculate the image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ .

The image of the line segment is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Conclusion

In conclusion, the reflection that will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ is a reflection across the axis of reflection ${y=-5x-26}$ .

Final Answer

The final answer is a reflection across the axis of reflection ${y=-5x-26}$ .

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,

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Introduction

In the previous article, we explored the concept of reflecting a line segment across a line and determined which reflection will produce an image with endpoints at [ $(4,-6)\$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$. In this article, we will answer some frequently asked questions about reflecting a line segment across a line.

Q&A

Q: What is the axis of reflection?

A: The axis of reflection is the line that passes through the midpoint of the line segment and is perpendicular to the line segment.

Q: How do I find the axis of reflection?

A: To find the axis of reflection, you need to find the midpoint of the line segment and the slope of the line segment. The midpoint is the average of the x-coordinates and the average of the y-coordinates of the endpoints. The slope is the change in y-coordinates divided by the change in x-coordinates.

Q: What is the equation of the axis of reflection?

A: The equation of the axis of reflection is ${y=-5x-26}$ .

Q: How do I reflect a line segment across the axis of reflection?

A: To reflect a line segment across the axis of reflection, you need to find the image of the line segment. The image of the line segment is the same distance and direction from the axis of reflection as the original line segment.

Q: What is the image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ ?

A: The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Q: What is the final answer?

A: The final answer is a reflection across the axis of reflection ${y=-5x-26}$ .

Q: What is the axis of reflection?

A: The axis of reflection is the line that passes through the midpoint of the line segment and is perpendicular to the line segment.

Q: How do I find the axis of reflection?

Q: What is the equation of the axis of reflection?

A: The equation of the axis of reflection is ${y=-5x-26}$ .

Q: How do I reflect a line segment across the axis of reflection?

Q: What is the image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ ?

A: The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Q: What is the final answer?

A: The final answer is a reflection across the axis of reflection ${y=-5x-26}$ .

Q: What is the axis of reflection?

A: The axis of reflection is the line that passes through the midpoint of the line segment and is perpendicular to the line segment.

Q: How do I find the axis of reflection?

Q: What is the equation of the axis of reflection?

A: The equation of the axis of reflection is ${y=-5x-26}$ .

Q: How do I reflect a line segment across the axis of reflection?

Q: What is the image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ ?

A: The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Q: What is the final answer?

A: The final answer is a reflection across the axis of reflection ${y=-5x-26}$ .

Q: What is the axis of reflection?

A: The axis of reflection is the line that passes through the midpoint of the line segment and is perpendicular to the line segment.

Q: How do I find the axis of reflection?

Q: What is the equation of the axis of reflection?

A: The equation of the axis of reflection is ${y=-5x-26}$ .

Q: How do I reflect a line segment across the axis of reflection?

Q: What is the image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ ?

A: The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Q: What is the final answer?

A: The final answer is a reflection across the axis of reflection ${y=-5x-26}$ .

Conclusion

In conclusion, reflecting a line segment across a line is a transformation that flips a line segment over a line. The axis of reflection is the line that passes through the midpoint of the line segment and is perpendicular to the line segment. To find the axis of reflection, you need to find the midpoint of the line segment and the slope of the line segment. The image of the line segment is the same distance and direction from the axis of reflection as the original line segment.

Final Answer

The final answer is a reflection across the axis of reflection ${y=-5x-26}$ .

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection ${y=-5x-26}$ is the line segment with endpoints at ${(4,-6)\$}$ and ${$ (6,4)$}$.

Reflection Across the Axis of Reflection

A reflection across the axis of reflection ${y=-5x-26}$ will produce an image with endpoints at ${$ (4,-6)$}$ and ${$ (6,4)$}$ given the original endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$.

Reflection Across the Axis of Reflection

The axis of reflection ${y=-5x-26}$ is the line that passes through the midpoint ${(-5,-1)}$ and has a slope of ${-5}$ .

Reflection Across the Axis of Reflection

The image of the line segment with endpoints at ${$ (-4,-6)$}$ and ${$ (-6,4)$}$ across the axis of reflection