Reflection of a Triangle in the Coordinate Plane

Oct 13, 2025 by magicalceylontours.com 138 views

Introduction

In geometry, a reflection is a transformation that flips a figure over a line. This line is called the axis of reflection. When a triangle is reflected over a line, the resulting triangle is called the reflection of the original triangle. In this article, we will discuss the reflection of a triangle in the coordinate plane.

What is a Reflection?

A reflection is a transformation that flips a figure over a line. This line is called the axis of reflection. The axis of reflection can be a horizontal line, a vertical line, or a diagonal line. When a figure is reflected over a line, the resulting figure is called the reflection of the original figure.

Reflection of a Point

To reflect a point over a line, we need to find the distance between the point and the line. The distance between the point and the line is called the perpendicular distance. The perpendicular distance is the shortest distance between the point and the line.

Let's consider a point $P(x, y)$ and a line $y = a$ . The perpendicular distance between the point and the line is given by the formula:

d = \frac{|y - a|}{\sqrt{1 + 1}}

The reflection of the point $P(x, y)$ over the line $y = a$ is given by the formula:

P^{\prime}(x, y) = (x, 2a - y)

Reflection of a Triangle

To reflect a triangle over a line, we need to reflect each vertex of the triangle over the line. The reflection of a triangle is called the reflection of the original triangle.

Let's consider a triangle $ABC$ with vertices $A(4, 1)$ , $B(6, 3)$ , and $C(2, 5)$ . The reflection of the triangle $ABC$ over the line $y = 2$ is given by the formula:

A^{\prime}(4, 4)

B^{\prime}(6, 1)

C^{\prime}(2, 3)

Reflection of a Triangle in the Coordinate Plane

To reflect a triangle in the coordinate plane, we need to reflect each vertex of the triangle over the line. The reflection of a triangle in the coordinate plane is called the reflection of the original triangle.

Let's consider a triangle $ABC$ with vertices $A(4, 1)$ , $B(6, 3)$ , and $C(2, 5)$ . The reflection of the triangle $ABC$ over the line $y = 2$ is given by the formula:

A^{\prime}(4, 4)

B^{\prime}(6, 1)

C^{\prime}(2, 3)

Properties of Reflection

The reflection of a triangle has several properties. Some of the properties of reflection are:

The reflection of a triangle is a triangle.
The reflection of a triangle is congruent to the original triangle.
The reflection of a triangle is symmetric to the original triangle.

Examples of Reflection

Let's consider some examples of reflection.

Example 1: Reflect the triangle $ABC$ $A BC$ over the line $y = 2$ $y = 2$ .
- The vertices of the triangle $ABC$ are $A(4, 1)$ , $B(6, 3)$ , and $C(2, 5)$ .
- The reflection of the triangle $ABC$ over the line $y = 2$ is given by the formula: $A^{\prime}(4, 4)$ $B^{\prime}(6, 1)$ $C^{\prime}(2, 3)$
Example 2: Reflect the triangle $ABC$ $A BC$ over the line $x = 3$ $x = 3$ .
- The vertices of the triangle $ABC$ are $A(4, 1)$ , $B(6, 3)$ , and $C(2, 5)$ .
- The reflection of the triangle $ABC$ over the line $x = 3$ is given by the formula: $A^{\prime}(2, 1)$ $B^{\prime}(4, 3)$ $C^{\prime}(6, 5)$

Conclusion

In this article, we discussed the reflection of a triangle in the coordinate plane. We learned that the reflection of a triangle is a triangle, and the reflection of a triangle is congruent to the original triangle. We also learned that the reflection of a triangle is symmetric to the original triangle. We considered some examples of reflection and learned how to reflect a triangle over a line.

References

[1] "Geometry" by Michael Artin
[2] "Introduction to Geometry" by H.S.M. Coxeter
[3] "Geometry: A Comprehensive Introduction" by Dan Pedoe

Glossary

Axis of reflection: The line over which a figure is reflected.
Perpendicular distance: The shortest distance between a point and a line.
Reflection: A transformation that flips a figure over a line.
Symmetric: A figure that is congruent to its reflection.
Reflection of a Triangle in the Coordinate Plane: Q&A =====================================================

Introduction

In our previous article, we discussed the reflection of a triangle in the coordinate plane. We learned that the reflection of a triangle is a triangle, and the reflection of a triangle is congruent to the original triangle. We also learned that the reflection of a triangle is symmetric to the original triangle. In this article, we will answer some frequently asked questions about the reflection of a triangle in the coordinate plane.

Q: What is the axis of reflection?

A: The axis of reflection is the line over which a figure is reflected.

Q: What is the perpendicular distance?

A: The perpendicular distance is the shortest distance between a point and a line.

Q: How do I find the reflection of a point over a line?

A: To find the reflection of a point over a line, you need to find the distance between the point and the line. The distance between the point and the line is called the perpendicular distance. The reflection of the point over the line is given by the formula:

P^{\prime}(x, y) = (x, 2a - y)

Q: How do I find the reflection of a triangle over a line?

A: To find the reflection of a triangle over a line, you need to reflect each vertex of the triangle over the line. The reflection of a triangle is called the reflection of the original triangle.

Q: What are the properties of reflection?

A: The reflection of a triangle has several properties. Some of the properties of reflection are:

The reflection of a triangle is a triangle.
The reflection of a triangle is congruent to the original triangle.
The reflection of a triangle is symmetric to the original triangle.

Q: Can I reflect a triangle over a horizontal line?

A: Yes, you can reflect a triangle over a horizontal line. The reflection of a triangle over a horizontal line is given by the formula:

A^{\prime}(x, 2a - y)

B^{\prime}(x, 2a - y)

C^{\prime}(x, 2a - y)

Q: Can I reflect a triangle over a vertical line?

A: Yes, you can reflect a triangle over a vertical line. The reflection of a triangle over a vertical line is given by the formula:

A^{\prime}(2x - a, y)

B^{\prime}(2x - a, y)

C^{\prime}(2x - a, y)

Q: Can I reflect a triangle over a diagonal line?

A: Yes, you can reflect a triangle over a diagonal line. The reflection of a triangle over a diagonal line is given by the formula:

A^{\prime}(2x - a, 2y - b)

B^{\prime}(2x - a, 2y - b)

C^{\prime}(2x - a, 2y - b)

Q: What are some examples of reflection?

A: Some examples of reflection are:

Example 1: Reflect the triangle $ABC$ $A BC$ over the line $y = 2$ $y = 2$ .
- The vertices of the triangle $ABC$ are $A(4, 1)$ , $B(6, 3)$ , and $C(2, 5)$ .
- The reflection of the triangle $ABC$ over the line $y = 2$ is given by the formula: $A^{\prime}(4, 4)$ $B^{\prime}(6, 1)$ $C^{\prime}(2, 3)$
Example 2: Reflect the triangle $ABC$ $A BC$ over the line $x = 3$ $x = 3$ .
- The vertices of the triangle $ABC$ are $A(4, 1)$ , $B(6, 3)$ , and $C(2, 5)$ .
- The reflection of the triangle $ABC$ over the line $x = 3$ is given by the formula: $A^{\prime}(2, 1)$ $B^{\prime}(4, 3)$ $C^{\prime}(6, 5)$

Conclusion

In this article, we answered some frequently asked questions about the reflection of a triangle in the coordinate plane. We learned that the reflection of a triangle is a triangle, and the reflection of a triangle is congruent to the original triangle. We also learned that the reflection of a triangle is symmetric to the original triangle. We considered some examples of reflection and learned how to reflect a triangle over a line.

References

[1] "Geometry" by Michael Artin
[2] "Introduction to Geometry" by H.S.M. Coxeter
[3] "Geometry: A Comprehensive Introduction" by Dan Pedoe

Glossary

Axis of reflection: The line over which a figure is reflected.
Perpendicular distance: The shortest distance between a point and a line.
Reflection: A transformation that flips a figure over a line.
Symmetric: A figure that is congruent to its reflection.