Square { ABCD$}$ 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

Introduction

In geometry, coordinate translations are a fundamental concept used to describe the movement of points and shapes in a two-dimensional plane. A translation is a transformation that moves every point of a figure by the same distance in the same direction. In this article, we will explore how to apply coordinate translations to find the pre-image of a given point in a square.

The Translation Rule

The translation rule given is {(x, y) \rightarrow (x-4, y+15)$}$. This means that for any point {(x, y)$}$ in the original square, its image after the translation will be {(x-4, y+15)$}$.

Finding the Pre-Image of Point {D^{\prime}$]

We are given that the square [ABCD\$} was translated to form {A^{\prime}B{\prime}C^{\prime}D{\prime}$}$. To find the coordinates of point {D$}$ in the pre-image, we need to work backwards from the translated point {D^{\prime}$].

Let's assume that the coordinates of point [$D^{\prime}\$] are \[$(x^{\prime}, y^{\prime})$}$. Using the translation rule, we can write:

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

To find the coordinates of point {D$}$ in the pre-image, we need to isolate {x$}$ and {y$}$ in the above equation.

Isolating {x$}$ and {y$}$

We can start by isolating {x$}$ in the equation:

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

Adding 4 to both sides of the equation, we get:

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

Next, we can isolate {y$}$ in the equation:

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

Subtracting 15 from both sides of the equation, we get:

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

Finding the Coordinates of Point {D$}$

Now that we have isolated {x$}$ and {y$}$, we can substitute the coordinates of point {D^{\prime}$] into the equations:

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

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

Since we don't know the coordinates of point {D^{\prime}$], we will leave them as variables [$x^{\prime}\$] and \[$y^{\prime}$].

Conclusion

In this article, we explored how to apply coordinate translations to find the pre-image of a given point in a square. We used the translation rule [(x, y) \rightarrow (x-4, y+15)\$} to find the coordinates of point {D$}$ in the pre-image. By isolating {x$}$ and {y$}$ in the equation, we were able to express the coordinates of point {D$}$ in terms of the coordinates of point {D^{\prime}$].

Example

Suppose we know that the coordinates of point [$D^{\prime}\$] are \[$(10, 20)$}$. We can substitute these values into the equations to find the coordinates of point {D$}$:

{x = 10 + 4 = 14$}$

{y = 20 - 15 = 5$}$

Therefore, the coordinates of point {D$}$ in the pre-image are {(14, 5)$}$.

Applications of Coordinate Translations

Coordinate translations have many applications in geometry and other fields. Some examples include:

• Computer graphics: Coordinate translations are used to move objects in a 2D or 3D space.
• Game development: Coordinate translations are used to move game objects and characters.
• Architecture: Coordinate translations are used to design and build buildings and other structures.
• Engineering: Coordinate translations are used to design and build machines and other mechanical systems.

Conclusion

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Q: What is a coordinate translation?

A: A coordinate translation is a transformation that moves every point of a figure by the same distance in the same direction. It is a way of changing the position of a point or a shape in a two-dimensional plane.

Q: How do I apply a coordinate translation to a point?

A: To apply a coordinate translation to a point, you need to add or subtract a certain value from the x-coordinate and the y-coordinate of the point. The translation rule is given by [(x, y) \rightarrow (x + a, y + b)\$}, where {a$}$ and {b$}$ are the values to be added to the x-coordinate and the y-coordinate, respectively.

Q: What is the difference between a translation and a rotation?

A: A translation is a transformation that moves every point of a figure by the same distance in the same direction, whereas a rotation is a transformation that turns a figure around a fixed point. In a translation, the size and shape of the figure remain the same, but in a rotation, the size and shape of the figure may change.

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

A: To find the pre-image of a point after a translation, you need to work backwards from the translated point. You can do this by subtracting the translation values from the x-coordinate and the y-coordinate of the translated point.

Q: What are some real-world applications of coordinate translations?

A: Coordinate translations have many real-world applications, including:

• Computer graphics: Coordinate translations are used to move objects in a 2D or 3D space.
• Game development: Coordinate translations are used to move game objects and characters.
• Architecture: Coordinate translations are used to design and build buildings and other structures.
• Engineering: Coordinate translations are used to design and build machines and other mechanical systems.

Q: How do I determine the translation values in a coordinate translation?

A: The translation values in a coordinate translation are determined by the direction and distance of the translation. For example, if a point is translated 3 units to the right and 2 units up, the translation values are {a = 3$}$ and {b = 2$}$.

Q: Can I apply multiple translations to a point?

A: Yes, you can apply multiple translations to a point. To do this, you need to apply each translation in sequence, adding or subtracting the translation values from the x-coordinate and the y-coordinate of the point after each translation.

Q: How do I find the image of a point after multiple translations?

A: To find the image of a point after multiple translations, you need to apply each translation in sequence, adding or subtracting the translation values from the x-coordinate and the y-coordinate of the point after each translation.

Q: What is the difference between a translation and a reflection?

A: A translation is a transformation that moves every point of a figure by the same distance in the same direction, whereas a reflection is a transformation that flips a figure over a line or a point. In a translation, the size and shape of the figure remain the same, but in a reflection, the size and shape of the figure may change.

Q: Can I apply a translation to a shape with multiple points?

A: Yes, you can apply a translation to a shape with multiple points. To do this, you need to apply the translation to each point of the shape, adding or subtracting the translation values from the x-coordinate and the y-coordinate of each point.

Q: How do I find the pre-image of a shape after a translation?

A: To find the pre-image of a shape after a translation, you need to work backwards from the translated shape. You can do this by subtracting the translation values from the x-coordinate and the y-coordinate of each point of the translated shape.

Q: What are some common mistakes to avoid when applying coordinate translations?

A: Some common mistakes to avoid when applying coordinate translations include:

• Not applying the translation values correctly: Make sure to add or subtract the translation values from the x-coordinate and the y-coordinate of each point.
• Not working backwards from the translated point: Make sure to work backwards from the translated point to find the pre-image of the point.
• Not considering the direction of the translation: Make sure to consider the direction of the translation when applying the translation values.

Conclusion

In conclusion, coordinate translations are a fundamental concept in geometry that can be used to find the pre-image of a given point in a square. By applying the translation rule and isolating {x$}$ and {y$}$, we can express the coordinates of point {D$}$ in terms of the coordinates of point [$D^{\prime}$]. Coordinate translations have many applications in various fields, including computer graphics, game development, architecture, and engineering.