The Image Of A Point Is Given By The Rule R Y = − X ( X , Y ) → ( − 4 , 9 R_{y=-x}(x, Y) \rightarrow (-4, 9 R Y = − X ​ ( X , Y ) → ( − 4 , 9 ]. What Are The Coordinates?A. ( − 9 , 4 (-9, 4 ( − 9 , 4 ] B. ( − 4 , − 9 (-4, -9 ( − 4 , − 9 ] C. ( 4 , 9 (4, 9 ( 4 , 9 ] D. ( 9 , − 4 (9, -4 ( 9 , − 4 ]

Understanding Reflection Across a Line

In mathematics, a reflection across a line is a transformation that flips a point or a shape over a given line. This type of transformation is an example of a rigid motion, which preserves the size and shape of the original figure. When reflecting a point across a line, the resulting image is a new point that is the same distance from the line as the original point, but on the opposite side.

The Given Reflection Rule

The reflection rule given in the problem is $r_{y=-x}(x, y) \rightarrow (-4, 9)$. This rule indicates that the point $(x, y)$ is reflected across the line $y = -x$ to produce the image $(-4, 9)$.

Understanding the Line of Reflection

The line of reflection is the line across which the point is being reflected. In this case, the line of reflection is $y = -x$. This line has a slope of $-1$ and passes through the origin $(0, 0)$.

Finding the Coordinates of the Image

To find the coordinates of the image, we need to use the reflection rule. The reflection rule states that the image of a point $(x, y)$ reflected across the line $y = -x$ is given by $(-y, -x)$.

Applying the Reflection Rule

Using the reflection rule, we can find the coordinates of the image of the point $(x, y)$ reflected across the line $y = -x$. The image of the point $(x, y)$ is given by $(-y, -x)$.

Finding the Coordinates of the Image of the Given Point

Now, let's find the coordinates of the image of the point $(-4, 9)$ reflected across the line $y = -x$. Using the reflection rule, we can find the coordinates of the image by substituting $x = -4$ and $y = 9$ into the reflection rule.

Substituting the Values

Substituting $x = -4$ and $y = 9$ into the reflection rule, we get:

$(-y, -x) = (-9, -(-4))$

Simplifying the Expression

Simplifying the expression, we get:

$(-y, -x) = (-9, 4)$

Conclusion

Therefore, the coordinates of the image of the point $(-4, 9)$ reflected across the line $y = -x$ are $(-9, 4)$.

Answer

The correct answer is:

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about reflection across a line.

Q: What is reflection across a line?

A: Reflection across a line is a transformation that flips a point or a shape over a given line. This type of transformation is an example of a rigid motion, which preserves the size and shape of the original figure.

Q: What is the line of reflection?

A: The line of reflection is the line across which the point is being reflected. In the case of the reflection rule $r_{y=-x}(x, y) \rightarrow (-4, 9)$, the line of reflection is $y = -x$.

Q: How do I find the coordinates of the image of a point reflected across a line?

A: To find the coordinates of the image of a point reflected across a line, you can use the reflection rule. The reflection rule states that the image of a point $(x, y)$ reflected across the line $y = -x$ is given by $(-y, -x)$.

Q: What is the reflection rule?

A: The reflection rule is a formula that is used to find the coordinates of the image of a point reflected across a line. The reflection rule is given by $(-y, -x)$.

Q: How do I apply the reflection rule?

A: To apply the reflection rule, you need to substitute the values of $x$ and $y$ into the formula. For example, if you want to find the coordinates of the image of the point $(-4, 9)$ reflected across the line $y = -x$, you would substitute $x = -4$ and $y = 9$ into the formula.

Q: What is the difference between reflection and rotation?

A: Reflection and rotation are two different types of transformations. Reflection is a transformation that flips a point or a shape over a given line, while rotation is a transformation that turns a point or a shape around a given point.

Q: Can I reflect a point across a line that is not $y = -x$?

A: Yes, you can reflect a point across any line. The reflection rule will be different for each line, but the concept of reflection remains the same.

Q: How do I find the coordinates of the image of a point reflected across a line that is not $y = -x$?

A: To find the coordinates of the image of a point reflected across a line that is not $y = -x$, you need to use the equation of the line to find the coordinates of the image. This may involve using algebraic manipulations or geometric reasoning.

Q: What are some real-world applications of reflection across a line?

A: Reflection across a line has many real-world applications, including:

• Mirrors and reflection in optics
• Symmetry in art and design
• Reflection in physics and engineering
• Computer graphics and animation

Conclusion

In this article, we have answered some of the most frequently asked questions about reflection across a line. We hope that this article has provided you with a better understanding of this important concept in mathematics.