A Triangle On A Coordinate Plane Is Translated According To The Rule $T_{-3,5}(x, Y$\]. Which Is Another Way To Write This Rule?A. $(x, Y) \rightarrow (x-3, Y+5$\] B. $(x, Y) \rightarrow (x-3, Y-5$\] C. $(x, Y) \rightarrow

A Triangle on a Coordinate Plane: Understanding Translation Rules

Introduction

In mathematics, particularly in geometry and coordinate geometry, translation is a fundamental concept that involves moving a point or a shape from one location to another without changing its size or orientation. A translation rule is a mathematical expression that describes how to move a point or a shape from its original position to a new position. In this article, we will explore the concept of translation rules, specifically the rule $T_{-3,5}(x, y)$, and discuss how to write it in another way.

Understanding the Translation Rule $T_{-3,5}(x, y)$

The translation rule $T_{-3,5}(x, y)$ indicates that a point $(x, y)$ is translated to a new position by moving it $-3$ units horizontally (left) and $5$ units vertically (up). This can be represented graphically as follows:

• The original point $(x, y)$ is represented by a red dot.
• The translated point $(x-3, y+5)$ is represented by a blue dot.

Another Way to Write the Translation Rule

Now, let's consider the options provided to write the translation rule $T_{-3,5}(x, y)$ in another way.

• Option A: $(x, y) \rightarrow (x-3, y+5)$
• Option B: $(x, y) \rightarrow (x-3, y-5)$
• Option C: $(x, y) \rightarrow (x+3, y-5)$

To determine the correct answer, we need to analyze each option and compare it with the original translation rule $T_{-3,5}(x, y)$.

Analyzing Option A

Option A states that the translation rule is $(x, y) \rightarrow (x-3, y+5)$. This means that the point $(x, y)$ is translated to a new position by moving it $-3$ units horizontally (left) and $5$ units vertically (up). This is consistent with the original translation rule $T_{-3,5}(x, y)$.

Analyzing Option B

Option B states that the translation rule is $(x, y) \rightarrow (x-3, y-5)$. This means that the point $(x, y)$ is translated to a new position by moving it $-3$ units horizontally (left) and $-5$ units vertically (down). This is not consistent with the original translation rule $T_{-3,5}(x, y)$, which specifies a vertical translation of $5$ units up.

Analyzing Option C

Option C states that the translation rule is $(x, y) \rightarrow (x+3, y-5)$. This means that the point $(x, y)$ is translated to a new position by moving it $3$ units horizontally (right) and $-5$ units vertically (down). This is not consistent with the original translation rule $T_{-3,5}(x, y)$, which specifies a horizontal translation of $-3$ units and a vertical translation of $5$ units up.

Conclusion

Based on the analysis, we can conclude that the correct answer is Option A: $(x, y) \rightarrow (x-3, y+5)$. This is because it accurately represents the translation rule $T_{-3,5}(x, y)$, which involves moving a point $(x, y)$ $-3$ units horizontally (left) and $5$ units vertically (up).

Example Problems

To reinforce the understanding of translation rules, let's consider some example problems.

• Problem 1: Apply the translation rule $T_{-3,5}(x, y)$ to the point $(2, 3)$.
• Problem 2: Apply the translation rule $T_{-3,5}(x, y)$ to the point $(4, 6)$.
• Problem 3: Apply the translation rule $T_{-3,5}(x, y)$ to the point $(1, 2)$.

Solutions

• Problem 1: Applying the translation rule $T_{-3,5}(x, y)$ to the point $(2, 3)$, we get $(2-3, 3+5) = (-1, 8)$.
• Problem 2: Applying the translation rule $T_{-3,5}(x, y)$ to the point $(4, 6)$, we get $(4-3, 6+5) = (1, 11)$.
• Problem 3: Applying the translation rule $T_{-3,5}(x, y)$ to the point $(1, 2)$, we get $(1-3, 2+5) = (-2, 7)$.

Conclusion

In conclusion, the translation rule $T_{-3,5}(x, y)$ can be written in another way as $(x, y) \rightarrow (x-3, y+5)$. This is because it accurately represents the translation of a point $(x, y)$ $-3$ units horizontally (left) and $5$ units vertically (up). By understanding and applying translation rules, we can solve problems involving the movement of points and shapes in a coordinate plane.
A Triangle on a Coordinate Plane: Understanding Translation Rules - Q&A

Introduction

In our previous article, we explored the concept of translation rules, specifically the rule $T_{-3,5}(x, y)$, and discussed how to write it in another way. In this article, we will address some frequently asked questions (FAQs) related to translation rules and provide detailed answers to help you better understand this concept.

Q&A

Q1: What is a translation rule?

A1: A translation rule is a mathematical expression that describes how to move a point or a shape from its original position to a new position without changing its size or orientation.

Q2: How do I apply a translation rule to a point?

A2: To apply a translation rule to a point, you need to substitute the coordinates of the point into the translation rule and perform the specified horizontal and vertical translations.

Q3: What is the difference between a translation rule and a transformation rule?

A3: A translation rule is a specific type of transformation rule that involves moving a point or a shape from its original position to a new position without changing its size or orientation. Other types of transformation rules include rotation, reflection, and dilation.

Q4: Can I apply multiple translation rules to a point?

A4: Yes, you can apply multiple translation rules to a point. However, you need to apply the rules in the correct order, making sure that the final translation is the result of all the previous translations.

Q5: How do I determine the correct order of translation rules?

A5: To determine the correct order of translation rules, you need to analyze the rules and identify the horizontal and vertical translations involved. You can then apply the rules in the order that makes the most sense, usually from left to right and top to bottom.

Q6: Can I apply a translation rule to a shape?

A6: Yes, you can apply a translation rule to a shape. However, you need to apply the rule to each vertex of the shape, making sure that the shape remains intact.

Q7: How do I graph a translated shape?

A7: To graph a translated shape, you need to apply the translation rule to the original shape and then plot the resulting shape on the coordinate plane.

Q8: Can I use a translation rule to solve a problem involving a real-world scenario?

A8: Yes, you can use a translation rule to solve a problem involving a real-world scenario. For example, you can use a translation rule to model the movement of an object in a coordinate plane.

Q9: How do I determine the distance between two points after applying a translation rule?

A9: To determine the distance between two points after applying a translation rule, you need to use the distance formula and substitute the coordinates of the translated points into the formula.

Q10: Can I use a translation rule to solve a problem involving a system of equations?

A10: Yes, you can use a translation rule to solve a problem involving a system of equations. For example, you can use a translation rule to model the movement of an object in a coordinate plane and then solve the resulting system of equations.

Conclusion

In conclusion, translation rules are an essential concept in mathematics, particularly in geometry and coordinate geometry. By understanding and applying translation rules, you can solve problems involving the movement of points and shapes in a coordinate plane. We hope that this Q&A article has provided you with a better understanding of translation rules and how to apply them to solve real-world problems.

Example Problems

To reinforce your understanding of translation rules, let's consider some example problems.

• Problem 1: Apply the translation rule $T_{-3,5}(x, y)$ to the point $(2, 3)$ and then determine the distance between the original point and the translated point.
• Problem 2: Apply the translation rule $T_{-3,5}(x, y)$ to the point $(4, 6)$ and then solve the resulting system of equations.
• Problem 3: Apply the translation rule $T_{-3,5}(x, y)$ to the point $(1, 2)$ and then graph the resulting shape on the coordinate plane.

Solutions

• Problem 1: Applying the translation rule $T_{-3,5}(x, y)$ to the point $(2, 3)$, we get $(2-3, 3+5) = (-1, 8)$. The distance between the original point $(2, 3)$ and the translated point $(-1, 8)$ is $\sqrt{(2-(-1))^2 + (3-8)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}$.
• Problem 2: Applying the translation rule $T_{-3,5}(x, y)$ to the point $(4, 6)$, we get $(4-3, 6+5) = (1, 11)$. The resulting system of equations is $x = 1$ and $y = 11$.
• Problem 3: Applying the translation rule $T_{-3,5}(x, y)$ to the point $(1, 2)$, we get $(1-3, 2+5) = (-2, 7)$. The resulting shape is a triangle with vertices at $(-2, 7)$, $(1, 2)$, and $(4, 6)$.

Conclusion

In conclusion, translation rules are a powerful tool for solving problems involving the movement of points and shapes in a coordinate plane. By understanding and applying translation rules, you can solve a wide range of problems, from simple geometry problems to complex real-world scenarios. We hope that this Q&A article has provided you with a better understanding of translation rules and how to apply them to solve real-world problems.