Square \[$A B C D\$\] Was Translated Using The Rule \[$(x, Y) \rightarrow (x-4, Y+15)\$\] To Form \[$A^{\prime} B^{\prime} C^{\prime} D^{\prime}\$\]. What Are The Coordinates Of Point \[$D\$\] In The Pre-image If The

In geometry, a translation is a transformation that moves a figure from one location to another without changing its size or orientation. The given problem involves translating a square $A B C D$ using the rule $(x, y) \rightarrow (x-4, y+15)$ to form $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$. We are asked to find the coordinates of point $D$ in the pre-image.

Translation Rule and Its Effect

The translation rule $(x, y) \rightarrow (x-4, y+15)$ indicates that each point $(x, y)$ in the original figure is moved to a new location $(x-4, y+15)$. This means that the $x$-coordinate of each point is decreased by 4, and the $y$-coordinate is increased by 15.

Finding the Coordinates of Point D

To find the coordinates of point $D$ in the pre-image, we need to work backwards from the coordinates of point $D^{\prime}$. Let's assume that the coordinates of point $D^{\prime}$ are $(x^{\prime}, y^{\prime})$. Since the translation rule is $(x, y) \rightarrow (x-4, y+15)$, we can write the following equations:

$$x^{\prime} = x - 4$$

$$y^{\prime} = y + 15$$

We are given that the coordinates of point $D^{\prime}$ are $(x^{\prime}, y^{\prime})$. We can substitute these values into the above equations to get:

$$x^{\prime} = x - 4$$

$$y^{\prime} = y + 15$$

Now, we need to find the values of $x$ and $y$ that satisfy these equations. We can do this by solving the system of equations.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.

Rearranging the first equation, we get:

$$x = x^{\prime} + 4$$

Substituting this expression for $x$ into the second equation, we get:

$$y^{\prime} = y + 15$$

Substituting $x^{\prime} + 4$ for $x$, we get:

$$y^{\prime} = (x^{\prime} + 4) + 15$$

Simplifying the equation, we get:

$$y^{\prime} = x^{\prime} + 19$$

Now, we can substitute the value of $y^{\prime}$ into the equation:

$$y^{\prime} = y + 15$$

Substituting $x^{\prime} + 19$ for $y^{\prime}$, we get:

$$x^{\prime} + 19 = y + 15$$

Simplifying the equation, we get:

$$x^{\prime} = y - 4$$

Now, we have two equations:

$$x^{\prime} = x - 4$$

$$x^{\prime} = y - 4$$

We can equate the two expressions for $x^{\prime}$ to get:

$$x - 4 = y - 4$$

Simplifying the equation, we get:

$$x = y$$

Now, we can substitute the value of $x$ into one of the original equations to find the value of $y$. Let's substitute $x$ into the equation:

$$x^{\prime} = x - 4$$

Substituting $y$ for $x$, we get:

$$x^{\prime} = y - 4$$

We are given that the coordinates of point $D^{\prime}$ are $(x^{\prime}, y^{\prime})$. We can substitute these values into the above equation to get:

$$y^{\prime} = y - 4$$

Now, we can substitute the value of $y^{\prime}$ into the equation:

$$y^{\prime} = y + 15$$

Substituting $y - 4$ for $y^{\prime}$, we get:

$$y - 4 = y + 15$$

Simplifying the equation, we get:

$$-4 = 15$$

This is a contradiction, which means that the system of equations has no solution. Therefore, we cannot find the coordinates of point $D$ in the pre-image.

Conclusion

In this problem, we were asked to find the coordinates of point $D$ in the pre-image after a translation. However, we found that the system of equations has no solution, which means that we cannot find the coordinates of point $D$ in the pre-image.

References

• [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
• [2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note

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Q: What is the translation rule used to translate the square $A B C D$ to form $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$?

A: The translation rule used is $(x, y) \rightarrow (x-4, y+15)$, which means that each point $(x, y)$ in the original figure is moved to a new location $(x-4, y+15)$.

Q: How does the translation rule affect the coordinates of the points in the original figure?

A: The translation rule decreases the $x$-coordinate of each point by 4 and increases the $y$-coordinate by 15.

Q: Can we find the coordinates of point $D$ in the pre-image using the translation rule?

A: Unfortunately, we cannot find the coordinates of point $D$ in the pre-image using the translation rule. The system of equations has no solution, which means that we cannot determine the coordinates of point $D$ in the pre-image.

Q: Why can't we find the coordinates of point $D$ in the pre-image?

A: We cannot find the coordinates of point $D$ in the pre-image because the problem statement does not provide enough information to solve the system of equations. The translation rule is not sufficient to determine the coordinates of point $D$ in the pre-image.

Q: What are some common mistakes to avoid when working with translation rules?

A: Some common mistakes to avoid when working with translation rules include:

• Not understanding the effect of the translation rule on the coordinates of the points in the original figure.
• Not using the correct translation rule to translate the figure.
• Not checking for any contradictions or inconsistencies in the system of equations.

Q: How can we use the translation rule to solve other problems involving translations?

A: We can use the translation rule to solve other problems involving translations by applying the rule to the original figure and then using the resulting figure to solve the problem. For example, we can use the translation rule to find the coordinates of other points in the pre-image or to determine the dimensions of the translated figure.

Q: What are some real-world applications of translation rules?

A: Translation rules have many real-world applications, including:

• Computer graphics: Translation rules are used to translate objects in 2D and 3D space.
• Architecture: Translation rules are used to translate buildings and other structures.
• Engineering: Translation rules are used to translate mechanical systems and other devices.

Q: Can you provide some examples of translation rules?

A: Yes, here are some examples of translation rules:

• $(x, y) \rightarrow (x+3, y-2)$
• $(x, y) \rightarrow (x-2, y+4)$
• $(x, y) \rightarrow (x+5, y-3)$

Q: How can we determine the correct translation rule to use in a given problem?

A: We can determine the correct translation rule to use in a given problem by analyzing the problem statement and identifying the type of translation required. For example, if the problem involves translating a figure to the left, we would use a translation rule that decreases the $x$-coordinate of each point.