Finding A Vertex Of The Pre-ImageGiven The Dilation Rule D_{O, \frac{1}{3}}(x, Y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right ] And The Image S ′ T ′ U ′ V ′ S'T'U'V' S ′ T ′ U ′ V ′ , What Are The Coordinates Of Vertex V Of The Pre-image?A. (0, 0) B.

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Understanding the Dilation Rule

A dilation is a transformation that changes the size of a figure. In this case, we are given the dilation rule DO,13(x,y)→(13x,13y)D_{O, \frac{1}{3}}(x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right). This rule indicates that for any point (x,y)(x, y), its image under the dilation is obtained by multiplying both the xx and yy coordinates by 13\frac{1}{3}.

What is a Pre-Image?

The pre-image of a point is the original point before it undergoes a transformation. In this case, we are given the image S′T′U′V′S'T'U'V' and we need to find the coordinates of vertex V of the pre-image.

Finding the Pre-Image of Vertex V

To find the pre-image of vertex V, we need to apply the inverse of the dilation rule. The inverse of the dilation rule DO,13(x,y)→(13x,13y)D_{O, \frac{1}{3}}(x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right) is given by DO,3(x,y)→(3x,3y)D_{O, 3}(x, y) \rightarrow (3x, 3y).

Applying the Inverse Dilation Rule

Let the coordinates of vertex V be (x,y)(x, y). Then, the image of V under the dilation rule is given by (13x,13y)\left(\frac{1}{3}x, \frac{1}{3}y\right). Since this image is equal to S′T′U′V′S'T'U'V', we can set up the following equation:

(13x,13y)=S′T′U′V′\left(\frac{1}{3}x, \frac{1}{3}y\right) = S'T'U'V'

Finding the Coordinates of Vertex V

To find the coordinates of vertex V, we need to solve the equation above. Since the image of V under the dilation rule is equal to S′T′U′V′S'T'U'V', we can set up the following system of equations:

13x=x′\frac{1}{3}x = x' 13y=y′\frac{1}{3}y = y'

where (x′,y′)(x', y') are the coordinates of S′T′U′V′S'T'U'V'.

Solving the System of Equations

To solve the system of equations above, we can multiply both sides of each equation by 3:

x=3x′x = 3x' y=3y′y = 3y'

Finding the Coordinates of Vertex V

Since we know that the coordinates of S′T′U′V′S'T'U'V' are (3, 6), we can substitute these values into the equations above:

x=3(3)=9x = 3(3) = 9 y=3(6)=18y = 3(6) = 18

Therefore, the coordinates of vertex V are (9, 18).

Conclusion

In this article, we have shown how to find the coordinates of a vertex of the pre-image given the dilation rule and the image. We have applied the inverse of the dilation rule to find the pre-image of vertex V and have solved the resulting system of equations to find the coordinates of vertex V.

Final Answer

The final answer is (9, 18).

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Frequently Asked Questions

Q: What is a dilation?

A: A dilation is a transformation that changes the size of a figure. In this case, we are given the dilation rule DO,13(x,y)→(13x,13y)D_{O, \frac{1}{3}}(x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right), which indicates that for any point (x,y)(x, y), its image under the dilation is obtained by multiplying both the xx and yy coordinates by 13\frac{1}{3}.

Q: What is the pre-image of a point?

A: The pre-image of a point is the original point before it undergoes a transformation. In this case, we are given the image S′T′U′V′S'T'U'V' and we need to find the coordinates of vertex V of the pre-image.

Q: How do I find the pre-image of a point?

A: To find the pre-image of a point, you need to apply the inverse of the dilation rule. The inverse of the dilation rule DO,13(x,y)→(13x,13y)D_{O, \frac{1}{3}}(x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right) is given by DO,3(x,y)→(3x,3y)D_{O, 3}(x, y) \rightarrow (3x, 3y).

Q: What is the inverse of a dilation?

A: The inverse of a dilation is a transformation that reverses the effect of the dilation. In this case, the inverse of the dilation rule DO,13(x,y)→(13x,13y)D_{O, \frac{1}{3}}(x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right) is given by DO,3(x,y)→(3x,3y)D_{O, 3}(x, y) \rightarrow (3x, 3y).

Q: How do I apply the inverse dilation rule?

A: To apply the inverse dilation rule, you need to multiply both the xx and yy coordinates of the image by 3.

Q: What are the coordinates of vertex V?

A: The coordinates of vertex V are (9, 18).

Additional Resources

Common Mistakes

  • Not applying the inverse dilation rule correctly
  • Not multiplying both the xx and yy coordinates by 3
  • Not using the correct dilation rule

Conclusion

In this article, we have answered some frequently asked questions about finding a vertex of the pre-image given the dilation rule and the image. We have also provided additional resources and common mistakes to help you better understand the concept.

Final Answer

The final answer is (9, 18).

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