Finding the Equation of the Line of Reflection

Oct 13, 2025 by miodeya.com 2 views

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Introduction

Reflection is a fundamental concept in geometry that involves flipping a shape or an object over a line. In this article, we will explore the coordinates of a triangle before and after reflection over a line. We will also derive the equation of the line of reflection.

The Original Triangle

The original triangle has coordinates $P (1,4), Q (3,6),$ and $R (5,2)$ . To understand the reflection, let's first visualize the triangle.

Visualizing the Triangle

The triangle $PQR$ has a base of length 4 units and a height of 2 units. The coordinates of the vertices are $P (1,4), Q (3,6),$ and $R (5,2)$ .

The Reflected Triangle

The triangle is reflected over a line, and its image coordinates are $P^{\prime}(-1,4), Q^{\prime}(-3,6),$ and $R^{\prime}(-5,2)$ . To find the equation of the line of reflection, we need to analyze the coordinates of the reflected triangle.

Analyzing the Reflected Triangle

The reflected triangle $P^{\prime}Q^{\prime}R^{\prime}$ has a base of length 4 units and a height of 2 units. The coordinates of the vertices are $P^{\prime}(-1,4), Q^{\prime}(-3,6),$ and $R^{\prime}(-5,2)$ .

Finding the Equation of the Line of Reflection

To find the equation of the line of reflection, we need to find the midpoint of the line segment joining $P$ and $P^{\prime}$ , $Q$ and $Q^{\prime}$ , and $R$ and $R^{\prime}$ .

Midpoints of the Line Segments

The midpoint of the line segment joining $P$ and $P^{\prime}$ is $\left(\frac{1-1}{2},\frac{4+4}{2}\right)=(0,4)$ . The midpoint of the line segment joining $Q$ and $Q^{\prime}$ is $\left(\frac{3-3}{2},\frac{6+6}{2}\right)=(0,6)$ . The midpoint of the line segment joining $R$ and $R^{\prime}$ is $\left(\frac{5-5}{2},\frac{2+2}{2}\right)=(0,2)$ .

The Equation of the Line of Reflection

Since the midpoints of the line segments joining $P$ and $P^{\prime}$ , $Q$ and $Q^{\prime}$ , and $R$ and $R^{\prime}$ are all the same, the equation of the line of reflection is a vertical line passing through the point $(0,4)$ .

The Equation of the Line

The equation of the line of reflection is $x=0$ .

Conclusion

In this article, we have explored the coordinates of a triangle before and after reflection over a line. We have also derived the equation of the line of reflection. The equation of the line of reflection is a vertical line passing through the point $(0,4)$ .

Key Takeaways

The coordinates of the original triangle are $P (1,4), Q (3,6),$ and $R (5,2)$ .
The coordinates of the reflected triangle are $P^{\prime}(-1,4), Q^{\prime}(-3,6),$ and $R^{\prime}(-5,2)$ .
The equation of the line of reflection is $x=0$ .

Future Work

In the future, we can explore other types of reflections, such as reflection over a horizontal line or a diagonal line. We can also investigate the properties of the reflected triangle, such as its area and perimeter.

References

[1] "Reflection" by Math Open Reference. Math Open Reference, 2022.
[2] "Geometry" by Khan Academy. Khan Academy, 2022.

Introduction

In our previous article, we explored the coordinates of a triangle before and after reflection over a line. We also derived the equation of the line of reflection. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is reflection in geometry?

A1: Reflection is a fundamental concept in geometry that involves flipping a shape or an object over a line.

Q2: What are the coordinates of the original triangle?

A2: The coordinates of the original triangle are $P (1,4), Q (3,6),$ and $R (5,2)$ .

Q3: What are the coordinates of the reflected triangle?

A3: The coordinates of the reflected triangle are $P^{\prime}(-1,4), Q^{\prime}(-3,6),$ and $R^{\prime}(-5,2)$ .

Q4: How do we find the equation of the line of reflection?

A4: To find the equation of the line of reflection, we need to find the midpoint of the line segment joining $P$ and $P^{\prime}$ , $Q$ and $Q^{\prime}$ , and $R$ and $R^{\prime}$ .

Q5: What is the equation of the line of reflection?

A5: The equation of the line of reflection is $x=0$ .

Q6: What are the key takeaways from this article?

A6: The key takeaways from this article are:

The coordinates of the original triangle are $P (1,4), Q (3,6),$ and $R (5,2)$ .
The coordinates of the reflected triangle are $P^{\prime}(-1,4), Q^{\prime}(-3,6),$ and $R^{\prime}(-5,2)$ .
The equation of the line of reflection is $x=0$ .

Q7: What are some future work ideas related to this topic?

A7: Some future work ideas related to this topic are:

Exploring other types of reflections, such as reflection over a horizontal line or a diagonal line.
Investigating the properties of the reflected triangle, such as its area and perimeter.

Q8: What are some references for further reading on this topic?

A8: Some references for further reading on this topic are:

[1] "Reflection" by Math Open Reference. Math Open Reference, 2022.
[2] "Geometry" by Khan Academy. Khan Academy, 2022.

Q9: What are some tags related to this topic?

A9: Some tags related to this topic are:

Reflection
Triangle
Line of Reflection
Geometry
Math

Conclusion

In this article, we have answered some frequently asked questions related to the coordinates of a triangle before and after reflection over a line. We have also provided some key takeaways, future work ideas, and references for further reading on this topic.

Key Takeaways

The coordinates of the original triangle are $P (1,4), Q (3,6),$ and $R (5,2)$ .
The coordinates of the reflected triangle are $P^{\prime}(-1,4), Q^{\prime}(-3,6),$ and $R^{\prime}(-5,2)$ .
The equation of the line of reflection is $x=0$ .

Future Work

References

[1] "Reflection" by Math Open Reference. Math Open Reference, 2022.
[2] "Geometry" by Khan Academy. Khan Academy, 2022.