A Triangle Is Drawn On The Coordinate Plane. It Is Translated 4 Units Right And 3 Units Down. Which Rule Describes The Translation?A. \[$(x, Y) \rightarrow (x+3, Y-4)\$\]B. \[$(x, Y) \rightarrow (x+3, Y+4)\$\]C. \[$(x, Y)

by ADMIN 222 views

Introduction

In mathematics, the coordinate plane is a fundamental concept used to represent points, lines, and shapes in a two-dimensional space. One of the essential operations performed on geometric figures in the coordinate plane is translation, which involves moving a shape from one location to another without changing its size or orientation. In this article, we will explore the concept of translation and determine which rule describes the translation of a triangle 4 units right and 3 units down on the coordinate plane.

Understanding Translation

Translation is a transformation that moves a shape from one location to another without changing its size, shape, or orientation. It is a type of rigid motion that preserves the properties of the shape. In the coordinate plane, translation can be represented by a vector, which is a mathematical object that has both magnitude and direction.

Translation Rules

There are two main rules that describe translation in the coordinate plane:

  • Rule 1: {(x, y) \rightarrow (x+3, y-4)$}$ This rule indicates that the x-coordinate is increased by 3 units, and the y-coordinate is decreased by 4 units.
  • Rule 2: {(x, y) \rightarrow (x+3, y+4)$}$ This rule indicates that the x-coordinate is increased by 3 units, and the y-coordinate is increased by 4 units.

Determining the Translation Rule

To determine which rule describes the translation of a triangle 4 units right and 3 units down, we need to analyze the movement of the triangle. When a shape is translated 4 units right, its x-coordinate increases by 4 units. Similarly, when a shape is translated 3 units down, its y-coordinate decreases by 3 units.

Applying the Translation Rules

Let's apply the two translation rules to a point (x, y) to see which one matches the given translation.

  • Rule 1: {(x, y) \rightarrow (x+3, y-4)$}$ Applying this rule to a point (x, y), we get (x+3, y-4).
  • Rule 2: {(x, y) \rightarrow (x+3, y+4)$}$ Applying this rule to a point (x, y), we get (x+3, y+4).

Conclusion

Based on the analysis, we can conclude that the translation rule that describes the movement of a triangle 4 units right and 3 units down is:

{(x, y) \rightarrow (x+3, y-4)$}$

This rule indicates that the x-coordinate is increased by 3 units, and the y-coordinate is decreased by 4 units, which matches the given translation.

Example

To illustrate this concept, let's consider an example. Suppose we have a triangle with vertices at (2, 5), (4, 5), and (3, 6). If we translate this triangle 4 units right and 3 units down, the new vertices will be:

  • (2+4, 5-3) = (6, 2)
  • (4+4, 5-3) = (8, 2)
  • (3+4, 6-3) = (7, 3)

Using the translation rule {(x, y) \rightarrow (x+3, y-4)$}$, we can verify that the new vertices match the expected values.

Conclusion

Introduction

In our previous article, we explored the concept of translation and determined which rule describes the translation of a triangle 4 units right and 3 units down on the coordinate plane. In this article, we will continue to delve deeper into the world of translation and answer some frequently asked questions.

Q&A Session

Q1: What is the difference between translation and rotation?

A1: Translation and rotation are two different types of transformations in the coordinate plane. Translation involves moving a shape from one location to another without changing its size or orientation, whereas rotation involves rotating a shape around a fixed point without changing its size or position.

Q2: How do I determine the translation rule for a given movement?

A2: To determine the translation rule, you need to analyze the movement of the shape. If the shape is moved 4 units right, its x-coordinate increases by 4 units. If the shape is moved 3 units down, its y-coordinate decreases by 3 units. You can then use the translation rule {(x, y) \rightarrow (x+3, y-4)$}$ to describe the movement.

Q3: Can I use the translation rule to find the new coordinates of a point?

A3: Yes, you can use the translation rule to find the new coordinates of a point. For example, if you have a point (x, y) and you want to translate it 4 units right and 3 units down, you can use the translation rule {(x, y) \rightarrow (x+3, y-4)$}$ to get the new coordinates (x+3, y-4).

Q4: How do I apply the translation rule to a set of points?

A4: To apply the translation rule to a set of points, you need to replace each x-coordinate with (x+3) and each y-coordinate with (y-4). For example, if you have a set of points {(2, 5), (4, 5), (3, 6)} and you want to translate them 4 units right and 3 units down, you can use the translation rule to get the new set of points {(2+3, 5-4), (4+3, 5-4), (3+3, 6-4)}.

Q5: Can I use the translation rule to solve problems involving geometric transformations?

A5: Yes, you can use the translation rule to solve problems involving geometric transformations. For example, if you have a triangle with vertices at (2, 5), (4, 5), and (3, 6) and you want to translate it 4 units right and 3 units down, you can use the translation rule to find the new vertices of the triangle.

Example Problems

Problem 1:

Translate the point (2, 5) 4 units right and 3 units down using the translation rule {(x, y) \rightarrow (x+3, y-4)$}$.

Solution:

Using the translation rule, we get (2+3, 5-4) = (5, 1).

Problem 2:

Translate the triangle with vertices at (2, 5), (4, 5), and (3, 6) 4 units right and 3 units down using the translation rule {(x, y) \rightarrow (x+3, y-4)$}$.

Solution:

Using the translation rule, we get the new vertices (2+3, 5-4) = (5, 1), (4+3, 5-4) = (7, 1), and (3+3, 6-4) = (6, 2).

Conclusion

In conclusion, the translation rule {(x, y) \rightarrow (x+3, y-4)$}$ describes the movement of a triangle 4 units right and 3 units down on the coordinate plane. We have also answered some frequently asked questions and provided example problems to illustrate the concept of translation. Understanding translation rules is essential in mathematics, as it helps us analyze and solve problems involving geometric transformations.