$\overline{ PL }$ Has Endpoints $P (4,-6)$ And $L (-2,1)$. The Segment Is Translated Using The Mapping $(x, Y) \rightarrow(x+5, Y)$. What Are The Coordinates Of $P ^{\prime}$ And $L

Introduction

In mathematics, a line segment is a part of a line that is bounded by two distinct points. The translation of a line segment is a fundamental concept in geometry, where the segment is moved from its original position to a new position without changing its shape or size. In this article, we will explore the translation of a line segment using the given mapping and find the coordinates of the translated points.

Understanding the Translation Mapping

The translation mapping is given by the equation $(x, y) \rightarrow (x+5, y)$. This means that for any point $(x, y)$ on the original line segment, the corresponding point on the translated line segment will be $(x+5, y)$. In other words, the x-coordinate of the translated point is 5 units to the right of the original point, while the y-coordinate remains the same.

Finding the Coordinates of $P^{\prime}$

The original point $P$ has coordinates $(4, -6)$. To find the coordinates of the translated point $P^{\prime}$, we apply the translation mapping:

$$P^{\prime} = (4+5, -6) = (9, -6)$$

Therefore, the coordinates of $P^{\prime}$ are $(9, -6)$.

Finding the Coordinates of $L^{\prime}$

The original point $L$ has coordinates $(-2, 1)$. To find the coordinates of the translated point $L^{\prime}$, we apply the translation mapping:

$$L^{\prime} = (-2+5, 1) = (3, 1)$$

Therefore, the coordinates of $L^{\prime}$ are $(3, 1)$.

Conclusion

In this article, we have explored the translation of a line segment using the given mapping. We have found the coordinates of the translated points $P^{\prime}$ and $L^{\prime}$ by applying the translation mapping to the original points $P$ and $L$. The translation mapping is a fundamental concept in geometry, and understanding it is essential for solving problems involving the translation of line segments.

Example Problems

1. A line segment has endpoints $A(2, 3)$ and $B(5, 6)$. The segment is translated using the mapping $(x, y) \rightarrow (x+2, y)$. Find the coordinates of $A^{\prime}$ and $B^{\prime}$.
2. A line segment has endpoints $C(-3, 4)$ and $D(1, -2)$. The segment is translated using the mapping $(x, y) \rightarrow (x-1, y)$. Find the coordinates of $C^{\prime}$ and $D^{\prime}$.

Solutions

1. Applying the translation mapping to the original points $A$ and $B$, we get:

$$A^{\prime} = (2+2, 3) = (4, 3)$$

$$B^{\prime} = (5+2, 6) = (7, 6)$$

Therefore, the coordinates of $A^{\prime}$ and $B^{\prime}$ are $(4, 3)$ and $(7, 6)$, respectively.

2. Applying the translation mapping to the original points $C$ and $D$, we get:

$$C^{\prime} = (-3-1, 4) = (-4, 4)$$

$$D^{\prime} = (1-1, -2) = (0, -2)$$

Therefore, the coordinates of $C^{\prime}$ and $D^{\prime}$ are $(-4, 4)$ and $(0, -2)$, respectively.

Final Thoughts

Introduction

In our previous article, we explored the translation of a line segment using the given mapping. We found the coordinates of the translated points $P^{\prime}$ and $L^{\prime}$ by applying the translation mapping to the original points $P$ and $L$. In this article, we will answer some frequently asked questions about the translation of a line segment.

Q: What is the translation mapping?

A: The translation mapping is a mathematical function that describes how to move a point from its original position to a new position. In the case of a line segment, the translation mapping is given by the equation $(x, y) \rightarrow (x+5, y)$, which means that the x-coordinate of the translated point is 5 units to the right of the original point, while the y-coordinate remains the same.

Q: How do I find the coordinates of the translated points?

A: To find the coordinates of the translated points, you need to apply the translation mapping to the original points. This involves adding the translation value to the x-coordinate of the original point, while keeping the y-coordinate the same.

Q: What if the translation mapping is not a simple addition?

A: If the translation mapping is not a simple addition, you may need to use more complex mathematical operations to find the coordinates of the translated points. For example, if the translation mapping is given by the equation $(x, y) \rightarrow (x+2y, y)$, you would need to use algebraic manipulations to find the coordinates of the translated points.

Q: Can I translate a line segment by more than one unit?

A: Yes, you can translate a line segment by more than one unit. For example, if the translation mapping is given by the equation $(x, y) \rightarrow (x+5, y)$, you can translate the line segment by 10 units by applying the mapping 10 times.

Q: What if I want to translate a line segment in a different direction?

A: If you want to translate a line segment in a different direction, you need to adjust the translation mapping accordingly. For example, if you want to translate a line segment 5 units up, you would use the translation mapping $(x, y) \rightarrow (x, y+5)$.

Q: Can I translate a line segment that is not a straight line?

A: Yes, you can translate a line segment that is not a straight line. However, you need to be careful when applying the translation mapping, as the resulting line segment may not be a straight line.

Q: What are some real-world applications of the translation of a line segment?

A: The translation of a line segment has numerous real-world applications, including:

• Computer graphics: Translating line segments is a fundamental operation in computer graphics, where it is used to move objects on the screen.
• Engineering: Translating line segments is used in engineering to design and analyze mechanical systems, such as bridges and buildings.
• Science: Translating line segments is used in science to model and analyze physical systems, such as the motion of objects in space.

Conclusion

In conclusion, the translation of a line segment is a fundamental concept in mathematics that has numerous real-world applications. By understanding the translation mapping and applying it to the original points, we can find the coordinates of the translated points. This concept is essential for solving problems involving the translation of line segments, and it has numerous applications in mathematics, science, and engineering.