A Line Segment Has Endpoints At \[$(-1,4)\$\] And \[$(4,1)\$\]. Which Reflection Will Produce An Image With Endpoints At \[$(-4,1)\$\] And \[$(-1,-4)\$\]?A. A Reflection Of The Line Segment Across The

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Introduction

In geometry, a line segment is a part of a line that is bounded by two distinct points. When we reflect a line segment across a line or a point, we create a new line segment that is a mirror image of the original line segment. In this problem, we are given two endpoints of a line segment and asked to find the reflection that will produce an image with new endpoints.

Given Information

The endpoints of the original line segment are given as {(-1,4)$}$ and {(4,1)$}$. We are asked to find the reflection that will produce an image with endpoints at {(-4,1)$}$ and {(-1,-4)$}$.

Reflection Types

There are several types of reflections that we can perform on a line segment. These include:

  • Reflection across a line: This type of reflection involves reflecting the line segment across a given line.
  • Reflection across a point: This type of reflection involves reflecting the line segment across a given point.
  • Reflection across a line segment: This type of reflection involves reflecting the line segment across a given line segment.

Reflection Across a Line

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

Step 1: Find the Midpoint

The midpoint of the line segment is given by the formula:

(x1+x22,y1+y22)\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

where (x1,y1){(x_1,y_1)} and (x2,y2){(x_2,y_2)} are the endpoints of the line segment.

Plugging in the values, we get:

(−1+42,4+12)=(32,52)\left(\frac{-1+4}{2},\frac{4+1}{2}\right)=\left(\frac{3}{2},\frac{5}{2}\right)

Step 2: Find the Slope

The slope of the line segment is given by the formula:

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

Plugging in the values, we get:

m=1−44−(−1)=−35m=\frac{1-4}{4-(-1)}=\frac{-3}{5}

Step 3: Find the Perpendicular Bisector

The perpendicular bisector is a line that passes through the midpoint and is perpendicular to the line segment. The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment.

The slope of the perpendicular bisector is:

m′=−1m=−1−35=53m'=-\frac{1}{m}=-\frac{1}{-\frac{3}{5}}=\frac{5}{3}

The equation of the perpendicular bisector is:

y−y1=m′(x−x1)y-y_1=m'(x-x_1)

Plugging in the values, we get:

y−52=53(x−32)y-\frac{5}{2}=\frac{5}{3}\left(x-\frac{3}{2}\right)

Reflection Across a Point

To reflect a line segment across a point, we need to find the line that passes through the point and is perpendicular to the line segment.

Step 1: Find the Line

The line that passes through the point and is perpendicular to the line segment is given by the equation:

y−y1=m′(x−x1)y-y_1=m'(x-x_1)

where (x1,y1){(x_1,y_1)} is the point and m′{m'} is the slope of the perpendicular bisector.

Plugging in the values, we get:

y−1=53(x−(−1))y-1=\frac{5}{3}\left(x-(-1)\right)

Step 2: Find the Reflection

The reflection of the line segment across the point is given by the equation:

y−y1=m′(x−x1)y-y_1=m'(x-x_1)

where (x1,y1){(x_1,y_1)} is the point and m′{m'} is the slope of the perpendicular bisector.

Plugging in the values, we get:

y−1=53(x−(−1))y-1=\frac{5}{3}\left(x-(-1)\right)

Reflection Across a Line Segment

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

Step 1: Find the Midpoint

The midpoint of the line segment is given by the formula:

(x1+x22,y1+y22)\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

where (x1,y1){(x_1,y_1)} and (x2,y2){(x_2,y_2)} are the endpoints of the line segment.

Plugging in the values, we get:

(−1+42,4+12)=(32,52)\left(\frac{-1+4}{2},\frac{4+1}{2}\right)=\left(\frac{3}{2},\frac{5}{2}\right)

Step 2: Find the Slope

The slope of the line segment is given by the formula:

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

Plugging in the values, we get:

m=1−44−(−1)=−35m=\frac{1-4}{4-(-1)}=\frac{-3}{5}

Step 3: Find the Perpendicular Bisector

The perpendicular bisector is a line that passes through the midpoint and is perpendicular to the line segment. The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment.

The slope of the perpendicular bisector is:

m′=−1m=−1−35=53m'=-\frac{1}{m}=-\frac{1}{-\frac{3}{5}}=\frac{5}{3}

The equation of the perpendicular bisector is:

y−y1=m′(x−x1)y-y_1=m'(x-x_1)

Plugging in the values, we get:

y−52=53(x−32)y-\frac{5}{2}=\frac{5}{3}\left(x-\frac{3}{2}\right)

Conclusion

In this problem, we were asked to find the reflection that will produce an image with endpoints at {(-4,1)$}$ and {(-1,-4)$}$ given the endpoints of the original line segment as {(-1,4)$}$ and {(4,1)$}$. We found that the reflection across the line segment is the correct reflection.

Reflection Across the Line Segment

The reflection across the line segment is given by the equation:

y−y1=m′(x−x1)y-y_1=m'(x-x_1)

where (x1,y1){(x_1,y_1)} is the midpoint of the line segment and m′{m'} is the slope of the perpendicular bisector.

Plugging in the values, we get:

y−52=53(x−32)y-\frac{5}{2}=\frac{5}{3}\left(x-\frac{3}{2}\right)

This is the equation of the reflection across the line segment.

Final Answer

Introduction

In our previous article, we discussed the problem of finding the reflection that will produce an image with endpoints at {(-4,1)$}$ and {(-1,-4)$}$ given the endpoints of the original line segment as {(-1,4)$}$ and {(4,1)$}$. We found that the reflection across the line segment is the correct reflection.

Q&A

Q: What is a line segment?

A: A line segment is a part of a line that is bounded by two distinct points.

Q: What is a reflection?

A: A reflection is a transformation that flips a figure over a line or a point.

Q: What are the different types of reflections?

A: There are several types of reflections, including:

  • Reflection across a line: This type of reflection involves reflecting the line segment across a given line.
  • Reflection across a point: This type of reflection involves reflecting the line segment across a given point.
  • Reflection across a line segment: This type of reflection involves reflecting the line segment across a given line segment.

Q: How do I find the reflection across a line?

A: To find the reflection across a line, you need to find the perpendicular bisector of the line segment. The perpendicular bisector is a line that passes through the midpoint of the line segment and is perpendicular to the line segment.

Q: How do I find the reflection across a point?

A: To find the reflection across a point, you need to find the line that passes through the point and is perpendicular to the line segment.

Q: How do I find the reflection across a line segment?

A: To find the reflection across a line segment, you need to find the line that passes through the midpoint of the line segment and is perpendicular to the line segment.

Q: What is the equation of the reflection across a line segment?

A: The equation of the reflection across a line segment is given by the formula:

y−y1=m′(x−x1)y-y_1=m'(x-x_1)

where (x1,y1){(x_1,y_1)} is the midpoint of the line segment and m′{m'} is the slope of the perpendicular bisector.

Q: How do I find the midpoint of a line segment?

A: To find the midpoint of a line segment, you need to use the formula:

(x1+x22,y1+y22)\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

where (x1,y1){(x_1,y_1)} and (x2,y2){(x_2,y_2)} are the endpoints of the line segment.

Q: How do I find the slope of a line segment?

A: To find the slope of a line segment, you need to use the formula:

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

where (x1,y1){(x_1,y_1)} and (x2,y2){(x_2,y_2)} are the endpoints of the line segment.

Q: What is the slope of the perpendicular bisector?

A: The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment.

Q: How do I find the equation of the perpendicular bisector?

A: To find the equation of the perpendicular bisector, you need to use the formula:

y−y1=m′(x−x1)y-y_1=m'(x-x_1)

where (x1,y1){(x_1,y_1)} is the midpoint of the line segment and m′{m'} is the slope of the perpendicular bisector.

Conclusion

In this article, we answered some common questions related to the problem of finding the reflection that will produce an image with endpoints at {(-4,1)$}$ and {(-1,-4)$}$ given the endpoints of the original line segment as {(-1,4)$}$ and {(4,1)$}$. We hope that this article has been helpful in understanding the concept of reflection and how to find the reflection across a line segment.