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 $x$-axis B. A Reflection
Understanding Reflections in Mathematics
Reflections are a fundamental concept in mathematics, particularly in geometry and coordinate geometry. A reflection is a transformation that flips a point or a shape over a line, known as the axis of reflection. In this article, we will explore how to determine the reflection of a line segment across the coordinate plane.
The Problem
A line segment has endpoints at and . We need to find the reflection of this line segment that will produce an image with endpoints at and .
Step 1: Understanding the Concept of Reflection
To solve this problem, we need to understand the concept of reflection. A reflection is a transformation that flips a point or a shape over a line, known as the axis of reflection. In this case, we are looking for a reflection that will produce an image with endpoints at and .
Step 2: Identifying the Axis of Reflection
To find the axis of reflection, we need to identify the line that passes through the midpoint of the line segment and is perpendicular to the line segment. The midpoint of the line segment can be found by taking the average of the x-coordinates and the y-coordinates of the endpoints.
Step 3: Finding the Midpoint of the Line Segment
The midpoint of the line segment can be found by taking the average of the x-coordinates and the y-coordinates of the endpoints.
Midpoint = ((-4 + (-6))/2, (-6 + 4)/2) = (-5, -1)
Step 4: Finding the Slope of the Line Segment
The slope of the line segment can be found by using the formula:
m = (y2 - y1) / (x2 - x1)
m = (4 - (-6)) / (-6 - (-4)) = 10 / -2 = -5
Step 5: Finding the Equation of the Line Segment
The equation of the line segment can be found by using the point-slope form of a line:
y - y1 = m(x - x1)
y - (-6) = -5(x - (-4))
y + 6 = -5(x + 4)
y + 6 = -5x - 20
y = -5x - 26
Step 6: Finding the Axis of Reflection
The axis of reflection is a line that passes through the midpoint of the line segment and is perpendicular to the line segment. The slope of the axis of reflection is the negative reciprocal of the slope of the line segment.
m_axis = -1 / -5 = 1/5
The equation of the axis of reflection can be found by using the point-slope form of a line:
y - y1 = m_axis(x - x1)
y - (-1) = 1/5(x - (-5))
y + 1 = 1/5(x + 5)
y = 1/5(x + 5) - 1
Step 7: Finding the Reflection of the Line Segment
To find the reflection of the line segment, we need to find the image of the line segment under the reflection. The image of the line segment can be found by using the equation of the axis of reflection.
Step 8: Finding the Image of the Line Segment
The image of the line segment can be found by using the equation of the axis of reflection.
Image = (2(-5) - 4, 2(-1) - (-6))
Image = (-14, -2)
Step 9: Finding the Slope of the Image
The slope of the image can be found by using the formula:
m_image = (y2 - y1) / (x2 - x1)
m_image = (-2 - (-6)) / (-14 - (-4))
m_image = 4 / -10 = -2/5
Step 10: Finding the Equation of the Image
The equation of the image can be found by using the point-slope form of a line:
y - y1 = m_image(x - x1)
y - (-2) = -2/5(x - (-14))
y + 2 = -2/5(x + 14)
y = -2/5(x + 14) - 2
Step 11: Finding the Reflection of the Line Segment Across the x-axis
To find the reflection of the line segment across the x-axis, we need to find the image of the line segment under the reflection. The image of the line segment can be found by using the equation of the axis of reflection.
Step 12: Finding the Image of the Line Segment Across the x-axis
The image of the line segment across the x-axis can be found by using the equation of the axis of reflection.
Image = (2(-5) - 4, 2(-1) - (-6))
Image = (-14, -2)
Step 13: Finding the Slope of the Image Across the x-axis
The slope of the image across the x-axis can be found by using the formula:
m_image = (y2 - y1) / (x2 - x1)
m_image = (-2 - (-6)) / (-14 - (-4))
m_image = 4 / -10 = -2/5
Step 14: Finding the Equation of the Image Across the x-axis
The equation of the image across the x-axis can be found by using the point-slope form of a line:
y - y1 = m_image(x - x1)
y - (-2) = -2/5(x - (-14))
y + 2 = -2/5(x + 14)
y = -2/5(x + 14) - 2
Step 15: Conclusion
In conclusion, the reflection of the line segment across the x-axis is the image of the line segment under the reflection. The image of the line segment can be found by using the equation of the axis of reflection.
The final answer is: A. A reflection of the line segment across the x-axis
Understanding Reflections in Mathematics
Reflections are a fundamental concept in mathematics, particularly in geometry and coordinate geometry. A reflection is a transformation that flips a point or a shape over a line, known as the axis of reflection. In this article, we will explore the concept of reflection and answer some common questions related to it.
Q: What is a reflection in mathematics?
A: A reflection is a transformation that flips a point or a shape over a line, known as the axis of reflection.
Q: What is the axis of reflection?
A: The axis of reflection 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 axis of reflection?
A: To find the axis of reflection, you need to identify the line that passes through the midpoint of the line segment and is perpendicular to the line segment. The midpoint of the line segment can be found by taking the average of the x-coordinates and the y-coordinates of the endpoints.
Q: What is the midpoint of a line segment?
A: The midpoint of a line segment is the point that divides the line segment into two equal parts.
Q: How do I find the midpoint of a line segment?
A: To find the midpoint of a line segment, you need to take the average of the x-coordinates and the y-coordinates of the endpoints.
Q: What is the slope of a line segment?
A: The slope of a line segment is a measure of how steep the line segment is.
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 - y1) / (x2 - x1)
Q: What is the equation of a line segment?
A: The equation of a line segment is a mathematical expression that describes the line segment.
Q: How do I find the equation of a line segment?
A: To find the equation of a line segment, you need to use the point-slope form of a line:
y - y1 = m(x - x1)
Q: What is the reflection of a line segment across the x-axis?
A: The reflection of a line segment across the x-axis is the image of the line segment under the reflection.
Q: How do I find the reflection of a line segment across the x-axis?
A: To find the reflection of a line segment across the x-axis, you need to find the image of the line segment under the reflection. The image of the line segment can be found by using the equation of the axis of reflection.
Q: What is the axis of reflection for a line segment across the x-axis?
A: The axis of reflection for a line segment across the x-axis is the x-axis itself.
Q: How do I find the equation of the axis of reflection for a line segment across the x-axis?
A: To find the equation of the axis of reflection for a line segment across the x-axis, you need to use the equation of the x-axis:
y = 0
Q: What is the reflection of a line segment across the y-axis?
A: The reflection of a line segment across the y-axis is the image of the line segment under the reflection.
Q: How do I find the reflection of a line segment across the y-axis?
A: To find the reflection of a line segment across the y-axis, you need to find the image of the line segment under the reflection. The image of the line segment can be found by using the equation of the axis of reflection.
Q: What is the axis of reflection for a line segment across the y-axis?
A: The axis of reflection for a line segment across the y-axis is the y-axis itself.
Q: How do I find the equation of the axis of reflection for a line segment across the y-axis?
A: To find the equation of the axis of reflection for a line segment across the y-axis, you need to use the equation of the y-axis:
x = 0
Q: What is the reflection of a line segment across a line?
A: The reflection of a line segment across a line is the image of the line segment under the reflection.
Q: How do I find the reflection of a line segment across a line?
A: To find the reflection of a line segment across a line, you need to find the image of the line segment under the reflection. The image of the line segment can be found by using the equation of the axis of reflection.
Q: What is the axis of reflection for a line segment across a line?
A: The axis of reflection for a line segment across a line is the line itself.
Q: How do I find the equation of the axis of reflection for a line segment across a line?
A: To find the equation of the axis of reflection for a line segment across a line, you need to use the equation of the line.
The final answer is: A reflection is a transformation that flips a point or a shape over a line, known as the axis of reflection.