A Triangle On The Coordinate Plane Has Vertices At \[$ A(4,1), B(1,5), \$\] And \[$ C(1,1) \$\]. The Triangle Is Reflected Across The \[$ X \$\]-axis To Form A Triangle With Vertices \[$ D(4,-1), E(1,-5), \$\] And

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Introduction

In mathematics, the concept of reflection is a fundamental idea that helps us understand how shapes and figures can be transformed on the coordinate plane. When a shape is reflected across a line, it creates a mirror image of the original shape on the other side of the line. In this article, we will explore the concept of reflection across the x-axis and apply it to a triangle with vertices at A(4,1), B(1,5), and C(1,1).

Reflection Across the X-Axis

The x-axis is a horizontal line that divides the coordinate plane into two parts: the upper half and the lower half. When a point is reflected across the x-axis, its y-coordinate changes sign, while its x-coordinate remains the same. In other words, if a point has coordinates (x, y), its reflection across the x-axis will have coordinates (x, -y).

The Original Triangle

The original triangle has vertices at A(4,1), B(1,5), and C(1,1). To find the coordinates of the reflected triangle, we need to apply the reflection formula to each vertex.

  • Vertex A(4,1) will be reflected to D(4,-1)
  • Vertex B(1,5) will be reflected to E(1,-5)
  • Vertex C(1,1) will be reflected to F(1,-1)

The Reflected Triangle

The reflected triangle has vertices at D(4,-1), E(1,-5), and F(1,-1). We can plot these points on the coordinate plane to visualize the reflected triangle.

Properties of the Reflected Triangle

The reflected triangle has several properties that are similar to the original triangle. For example:

  • The x-coordinates of the vertices of the reflected triangle are the same as the x-coordinates of the vertices of the original triangle.
  • The y-coordinates of the vertices of the reflected triangle are the negative of the y-coordinates of the vertices of the original triangle.
  • The distance between the vertices of the reflected triangle is the same as the distance between the vertices of the original triangle.

Conclusion

In this article, we explored the concept of reflection across the x-axis and applied it to a triangle with vertices at A(4,1), B(1,5), and C(1,1). We found the coordinates of the reflected triangle and discussed its properties. The concept of reflection is an important idea in mathematics that helps us understand how shapes and figures can be transformed on the coordinate plane.

Reflection Across the X-Axis: Examples and Exercises

Example 1

Find the coordinates of the reflected triangle with vertices at A(2,3), B(4,2), and C(3,4) across the x-axis.

Solution

  • Vertex A(2,3) will be reflected to D(2,-3)
  • Vertex B(4,2) will be reflected to E(4,-2)
  • Vertex C(3,4) will be reflected to F(3,-4)

Example 2

Find the coordinates of the reflected triangle with vertices at A(1,2), B(3,1), and C(2,3) across the x-axis.

Solution

  • Vertex A(1,2) will be reflected to D(1,-2)
  • Vertex B(3,1) will be reflected to E(3,-1)
  • Vertex C(2,3) will be reflected to F(2,-3)

Reflection Across the X-Axis: Applications

The concept of reflection across the x-axis has several applications in mathematics and real-world problems. For example:

  • In geometry, reflection is used to create mirror images of shapes and figures.
  • In trigonometry, reflection is used to find the coordinates of points on the unit circle.
  • In physics, reflection is used to describe the behavior of light and other waves.

Reflection Across the X-Axis: Exercises

Exercise 1

Find the coordinates of the reflected triangle with vertices at A(5,2), B(3,4), and C(2,3) across the x-axis.

Exercise 2

Find the coordinates of the reflected triangle with vertices at A(1,1), B(2,3), and C(3,2) across the x-axis.

Exercise 3

Find the coordinates of the reflected triangle with vertices at A(4,5), B(2,3), and C(3,4) across the x-axis.

Reflection Across the X-Axis: Conclusion

Introduction

In our previous article, we explored the concept of reflection across the x-axis and applied it to a triangle with vertices at A(4,1), B(1,5), and C(1,1). We found the coordinates of the reflected triangle and discussed its properties. In this article, we will answer some frequently asked questions about reflection across the x-axis.

Q&A

Q: What is reflection across the x-axis?

A: Reflection across the x-axis is a transformation that flips a shape or figure over the x-axis. When a point is reflected across the x-axis, its y-coordinate changes sign, while its x-coordinate remains the same.

Q: How do I find the coordinates of the reflected triangle?

A: To find the coordinates of the reflected triangle, you need to apply the reflection formula to each vertex. If a point has coordinates (x, y), its reflection across the x-axis will have coordinates (x, -y).

Q: What are the properties of the reflected triangle?

A: The reflected triangle has several properties that are similar to the original triangle. For example:

  • The x-coordinates of the vertices of the reflected triangle are the same as the x-coordinates of the vertices of the original triangle.
  • The y-coordinates of the vertices of the reflected triangle are the negative of the y-coordinates of the vertices of the original triangle.
  • The distance between the vertices of the reflected triangle is the same as the distance between the vertices of the original triangle.

Q: Can I reflect a triangle across the x-axis more than once?

A: Yes, you can reflect a triangle across the x-axis more than once. Each reflection will create a new triangle with the same properties as the original triangle.

Q: How do I graph the reflected triangle?

A: To graph the reflected triangle, you need to plot the vertices of the reflected triangle on the coordinate plane. You can use a ruler or a graphing calculator to help you plot the vertices.

Q: What are some real-world applications of reflection across the x-axis?

A: Reflection across the x-axis has several real-world applications, including:

  • In geometry, reflection is used to create mirror images of shapes and figures.
  • In trigonometry, reflection is used to find the coordinates of points on the unit circle.
  • In physics, reflection is used to describe the behavior of light and other waves.

Q: Can I reflect a triangle across the x-axis using a formula?

A: Yes, you can reflect a triangle across the x-axis using a formula. The formula for reflection across the x-axis is:

(x, y) → (x, -y)

Where (x, y) is the original point and (x, -y) is the reflected point.

Q: How do I find the coordinates of the reflected triangle using a formula?

A: To find the coordinates of the reflected triangle using a formula, you need to apply the reflection formula to each vertex. For example, if the original triangle has vertices at A(4,1), B(1,5), and C(1,1), the reflected triangle will have vertices at D(4,-1), E(1,-5), and F(1,-1).

Conclusion

In conclusion, reflection across the x-axis is an important concept in mathematics that helps us understand how shapes and figures can be transformed on the coordinate plane. We answered some frequently asked questions about reflection across the x-axis and provided examples and exercises to help you practice.