A Line Segment Has Endpoints At (-4, -6) And (-6, 4). Which Reflection Will Produce An Image With Endpoints At (4, -6) And (6, 4)?A. A Reflection Of The Line Segment Across The X-axisB. A Reflection Of The Line Segment Across The Y-axisC. A Reflection
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Introduction
Reflections are an essential concept in mathematics, particularly in geometry and coordinate geometry. A reflection is a transformation that flips a figure over a line, known as the line of reflection. In this article, we will explore the concept of reflections 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).
Understanding Reflections
Reflections are a type of transformation that can be performed on a figure in the coordinate plane. There are two main types of reflections: reflection across the x-axis and reflection across the y-axis.
Reflection Across the X-Axis
A reflection across the x-axis is a transformation that flips a figure over the x-axis. This means that the y-coordinates of the endpoints of the figure will change sign, while the x-coordinates remain the same.
Reflection Across the Y-Axis
A reflection across the y-axis is a transformation that flips a figure over the y-axis. This means that the x-coordinates of the endpoints of the figure will change sign, while the y-coordinates remain the same.
Reflection Across the Coordinate Plane
In addition to reflections across the x-axis and y-axis, there are also reflections across other lines, such as the line y = x. However, in this article, we will focus on reflections across the x-axis and y-axis.
Problem Analysis
Given the original endpoints at (-4, -6) and (-6, 4), we need to determine which reflection will produce an image with endpoints at (4, -6) and (6, 4).
Step 1: Determine the Type of Reflection
To determine the type of reflection, we need to analyze the change in the x-coordinates and y-coordinates of the endpoints.
X-Coordinates
The x-coordinates of the original endpoints are -4 and -6. The x-coordinates of the image endpoints are 4 and 6. This indicates that the x-coordinates have changed sign, which is a characteristic of a reflection across the y-axis.
Y-Coordinates
The y-coordinates of the original endpoints are -6 and 4. The y-coordinates of the image endpoints are -6 and 4. This indicates that the y-coordinates have not changed sign, which is a characteristic of a reflection across the y-axis.
Step 2: Determine the Line of Reflection
Based on the analysis in Step 1, we can conclude that the reflection is across the y-axis.
Step 3: Determine the Image Endpoints
To determine the image endpoints, we need to apply the reflection across the y-axis to the original endpoints.
Reflection Across the Y-Axis
A reflection across the y-axis is a transformation that flips a figure over the y-axis. This means that the x-coordinates of the endpoints will change sign, while the y-coordinates remain the same.
Image Endpoints
Applying the reflection across the y-axis to the original endpoints, we get:
- (-4, -6) → (4, -6)
- (-6, 4) → (6, 4)
Conclusion
Based on the analysis, we can conclude that the reflection that will produce an image with endpoints at (, -6) and (6, 4) is a reflection across the y-axis.
Final Answer
The final answer is B. A reflection of the line segment across the y-axis.
Discussion
Reflections are an essential concept in mathematics, particularly in geometry and coordinate geometry. In this article, we explored the concept of reflections 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). We analyzed the change in the x-coordinates and y-coordinates of the endpoints and concluded that the reflection is across the y-axis. This article provides a comprehensive understanding of reflections and their applications in mathematics.
References
- [1] "Geometry" by Michael Artin
- [2] "Coordinate Geometry" by David Guichard
- [3] "Reflections" by Math Open Reference
Keywords
Reflections, coordinate plane, x-axis, y-axis, line of reflection, image endpoints, transformation, geometry, coordinate geometry.
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Introduction
In our previous article, we explored the concept of reflections 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 provide a Q&A section to further clarify the concept of reflections and their applications in mathematics.
Q&A
Q1: What is a reflection in mathematics?
A1: A reflection is a transformation that flips a figure over a line, known as the line of reflection. It is a type of isometry that preserves the size and shape of the figure.
Q2: What are the two main types of reflections?
A2: The two main types of reflections are:
- Reflection across the x-axis
- Reflection across the y-axis
Q3: What is the difference between a reflection across the x-axis and a reflection across the y-axis?
A3: A reflection across the x-axis flips a figure over the x-axis, changing the sign of the y-coordinates while keeping the x-coordinates the same. A reflection across the y-axis flips a figure over the y-axis, changing the sign of the x-coordinates while keeping the y-coordinates the same.
Q4: How do you determine the line of reflection?
A4: To determine the line of reflection, you need to analyze the change in the x-coordinates and y-coordinates of the endpoints. If the x-coordinates change sign, the line of reflection is the y-axis. If the y-coordinates change sign, the line of reflection is the x-axis.
Q5: What is the image of a point after a reflection across the y-axis?
A5: The image of a point (x, y) after a reflection across the y-axis is (-x, y).
Q6: What is the image of a point after a reflection across the x-axis?
A6: The image of a point (x, y) after a reflection across the x-axis is (x, -y).
Q7: Can a reflection be across a line other than the x-axis or y-axis?
A7: Yes, a reflection can be across a line other than the x-axis or y-axis. For example, a reflection across the line y = x is a type of reflection that swaps the x and y coordinates.
Q8: What is the importance of reflections in mathematics?
A8: Reflections are an essential concept in mathematics, particularly in geometry and coordinate geometry. They are used to solve problems involving transformations, congruence, and similarity.
Q9: How do you apply reflections to solve problems in mathematics?
A9: To apply reflections to solve problems in mathematics, you need to:
- Identify the type of reflection (across the x-axis or y-axis)
- Determine the line of reflection
- Apply the reflection to the original figure
- Analyze the resulting image
Q10: What are some real-world applications of reflections?
A10: Reflections have many real-world applications, including:
- Designing buildings and bridges
- Creating art and graphics
- Solving problems in physics and engineering
- Understanding the behavior of light and sound
Conclusion
In this article, we provided a Q&A section further clarify the concept of reflections and their applications in mathematics. We hope that this article has been helpful in understanding the importance of reflections in mathematics and their real-world applications.
Final Answer
The final answer is B. A reflection of the line segment across the y-axis.
Discussion
Reflections are an essential concept in mathematics, particularly in geometry and coordinate geometry. In this article, we explored the concept of reflections and their applications in mathematics. We provided a Q&A section to further clarify the concept of reflections and their importance in mathematics.
References
- [1] "Geometry" by Michael Artin
- [2] "Coordinate Geometry" by David Guichard
- [3] "Reflections" by Math Open Reference
Keywords
Reflections, coordinate plane, x-axis, y-axis, line of reflection, image endpoints, transformation, geometry, coordinate geometry.