Given The Dilation Rule $D_{O, 1 / 3}(x, Y) \rightarrow \left(\frac{1}{3} X, \frac{1}{3} Y\right$\] And The Image S'T'UV', What Are The Coordinates Of Vertex V Of The Pre-image?A. (0, 0) B. $(0, \frac{1}{3}$\] C. (0, 1) D. (0, 3)

Introduction to Dilation

Dilation is a transformation that changes the size of a figure. In this case, we are given the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$. This means that the original figure is being dilated by a scale factor of $\frac{1}{3}$ with respect to the origin $O$. The coordinates of the pre-image are being scaled down by a factor of $\frac{1}{3}$ in both the $x$ and $y$ directions.

Understanding the Image S'T'UV'

The image S'T'UV' is the result of applying the dilation rule to the pre-image. Since the dilation rule is $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$, we can see that the coordinates of the image are being scaled down by a factor of $\frac{1}{3}$ in both the $x$ and $y$ directions.

Finding the Coordinates of Vertex V

To find the coordinates of vertex V of the pre-image, we need to work backwards from the image S'T'UV'. Since the dilation rule is $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$, we can see that the coordinates of the pre-image are being scaled up by a factor of $3$ in both the $x$ and $y$ directions.

Let's assume that the coordinates of vertex V of the pre-image are $(x, y)$. Then, applying the dilation rule, we get:

$$\left(\frac{1}{3} x, \frac{1}{3} y\right) = \text{coordinates of vertex V of the image}$$

Since the coordinates of vertex V of the image are $(0, 1)$, we can set up the following equations:

$$\frac{1}{3} x = 0$$

$$\frac{1}{3} y = 1$$

Solving for $x$ and $y$, we get:

$$x = 0$$

$$y = 3$$

Therefore, the coordinates of vertex V of the pre-image are $(0, 3)$.

Conclusion

In this article, we have discussed the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ and the image S'T'UV'. We have also worked backwards from the image to find the coordinates of vertex V of the pre-image. The coordinates of vertex V of the pre-image are $(0, 3)$.

References

• [1] "Dilation" by Math Open Reference. Retrieved February 26, 2024, from https://www.mathopenref.com/dilation.html
• [2] "Dilation Rule" by Khan Academy. Retrieved February 26, 2024, from https://www.khanacademy.org/math/geometry/transformations/dilation/v/dilation-rule

Discussion

Q: What is dilation and how does it relate to the image S'T'UV'?

A: Dilation is a transformation that changes the size of a figure. In this case, the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ means that the original figure is being dilated by a scale factor of $\frac{1}{3}$ with respect to the origin $O$. The image S'T'UV' is the result of applying this dilation rule to the pre-image.

Q: How do I find the coordinates of vertex V of the pre-image?

A: To find the coordinates of vertex V of the pre-image, you need to work backwards from the image S'T'UV'. Since the dilation rule is $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$, you can see that the coordinates of the pre-image are being scaled up by a factor of $3$ in both the $x$ and $y$ directions. Let's assume that the coordinates of vertex V of the pre-image are $(x, y)$. Then, applying the dilation rule, you get:

$$\left(\frac{1}{3} x, \frac{1}{3} y\right) = \text{coordinates of vertex V of the image}$$

Since the coordinates of vertex V of the image are $(0, 1)$, you can set up the following equations:

$$\frac{1}{3} x = 0$$

$$\frac{1}{3} y = 1$$

Solving for $x$ and $y$, you get:

$$x = 0$$

$$y = 3$$

Therefore, the coordinates of vertex V of the pre-image are $(0, 3)$.

Q: What is the scale factor of the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$?

A: The scale factor of the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ is $\frac{1}{3}$. This means that the original figure is being dilated by a scale factor of $\frac{1}{3}$ with respect to the origin $O$.

Q: How does the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ relate to the image S'T'UV'?

A: The dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ is applied to the pre-image to get the image S'T'UV'. Since the dilation rule is $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$, the coordinates of the image are being scaled down by a factor of $\frac{1}{3}$ in both the $x$ and $y$ directions.

Q: What are the coordinates of vertex V of the pre-image?

A: The coordinates of vertex V of the pre-image are $(0, 3)$.

Q: How do I apply the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ to the pre-image?

A: To apply the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ to the pre-image, you need to multiply the coordinates of the pre-image by the scale factor of $\frac{1}{3}$. This will give you the coordinates of the image.

Conclusion

In this Q&A article, we have discussed the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ and its relation to the image S'T'UV'. We have also answered some common questions about the dilation rule and the coordinates of vertex V of the pre-image.

References

• [1] "Dilation" by Math Open Reference. Retrieved February 26, 2024, from https://www.mathopenref.com/dilation.html
• [2] "Dilation Rule" by Khan Academy. Retrieved February 26, 2024, from https://www.khanacademy.org/math/geometry/transformations/dilation/v/dilation-rule

Discussion

Do you have any questions about the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ or the image S'T'UV'?