Triangle ABC Is Translated According To The Rule { (x, Y) \rightarrow (x+2, Y-8)$}$. If The Coordinates Of The Pre-image Of Point B Are { (4,-5)$}$, What Are The Coordinates Of { B^{\prime}$}$?A. { (2,3)$}$ B.
Understanding the Translation Rule
When a point is translated according to a given rule, its coordinates change based on the specific transformation applied. In this case, the translation rule is {(x, y) \rightarrow (x+2, y-8)$}$, which means that for every point {(x, y)$}$, its new coordinates will be {(x+2, y-8)$}$.
Applying the Translation Rule to Point B
The coordinates of the pre-image of point B are given as {(4,-5)$}$. To find the coordinates of point B', we need to apply the translation rule to the coordinates of point B.
Step 1: Apply the x-translation
The x-coordinate of point B is 4. According to the translation rule, the new x-coordinate will be ${4+2=6\$}.
Step 2: Apply the y-translation
The y-coordinate of point B is -5. According to the translation rule, the new y-coordinate will be {-5-8=-13$}$.
Finding the Coordinates of Point B'
By applying the translation rule to the coordinates of point B, we have found the coordinates of point B' to be {(6,-13)$}$.
Conclusion
In this article, we have discussed the translation of a point according to a given rule. We have applied the translation rule to the coordinates of point B to find the coordinates of point B'. The coordinates of point B' are {(6,-13)$}$.
Example Problem
Suppose we have a point A with coordinates {(x, y)$}$ and a translation rule {(x, y) \rightarrow (x-3, y+4)$}$. If the coordinates of the pre-image of point A are {(2,1)$}$, what are the coordinates of {A^{\prime}$}$?
Solution
To find the coordinates of point A', we need to apply the translation rule to the coordinates of point A.
Step 1: Apply the x-translation
The x-coordinate of point A is 2. According to the translation rule, the new x-coordinate will be ${2-3=-1\$}.
Step 2: Apply the y-translation
The y-coordinate of point A is 1. According to the translation rule, the new y-coordinate will be ${1+4=5\$}.
Finding the Coordinates of Point A'
By applying the translation rule to the coordinates of point A, we have found the coordinates of point A' to be {(-1,5)$}$.
Real-World Applications
Translation is an important concept in mathematics and has many real-world applications. For example, in computer graphics, translation is used to move objects on a screen. In engineering, translation is used to design and build structures such as bridges and buildings.
Common Mistakes
When applying a translation rule, it's easy to get the coordinates mixed up. Make sure to apply the x-translation and y-translation separately and accurately.
Tips and Tricks
When working with translation rules, it's helpful to use a coordinate plane to visualize the transformation. This can help you understand how the coordinates change and make it easier to apply the translation rule.
Conclusion
In this article, we have discussed the translation of a point according to a given rule. We have applied the translation rule to the coordinates of point B to find the coordinates of point B'. We have also discussed the importance of translation in mathematics and its real-world applications. By following the steps outlined in this article, you can confidently apply translation rules to find the coordinates of points.
Frequently Asked Questions
Q: What is the translation rule?
A: The translation rule is a mathematical formula that describes how to move a point from one location to another. In this case, the translation rule is {(x, y) \rightarrow (x+2, y-8)$}$, which means that for every point {(x, y)$}$, its new coordinates will be {(x+2, y-8)$}$.
Q: How do I apply the translation rule to a point?
A: To apply the translation rule to a point, you need to add 2 to the x-coordinate and subtract 8 from the y-coordinate. For example, if the coordinates of the pre-image of point B are {(4,-5)$}$, the coordinates of point B' will be {(4+2, -5-8) = (6, -13)$}$.
Q: What is the difference between the pre-image and the image of a point?
A: The pre-image of a point is the original point before it is translated, while the image of a point is the new point after it is translated. In the example above, the pre-image of point B is {(4,-5)$}$ and the image of point B is {(6,-13)$}$.
Q: Can I apply the translation rule to a point with negative coordinates?
A: Yes, you can apply the translation rule to a point with negative coordinates. For example, if the coordinates of the pre-image of point A are {(-3,2)$}$, the coordinates of point A' will be {(-3+2, 2-8) = (-1, -6)$}$.
Q: Can I apply the translation rule to a point with decimal coordinates?
A: Yes, you can apply the translation rule to a point with decimal coordinates. For example, if the coordinates of the pre-image of point B are {(2.5,-3.2)$}$, the coordinates of point B' will be {(2.5+2, -3.2-8) = (4.5, -11.2)$}$.
Q: Can I apply the translation rule to a point with no coordinates?
A: No, you cannot apply the translation rule to a point with no coordinates. The translation rule requires two coordinates, x and y, to work.
Q: Can I apply the translation rule to a point in 3D space?
A: Yes, you can apply the translation rule to a point in 3D space. However, you will need to add 2 to the x-coordinate, subtract 8 from the y-coordinate, and add 0 to the z-coordinate.
Q: Can I apply the translation rule to a point in a different coordinate system?
A: Yes, you can apply the translation rule to a point in a different coordinate system. However, you will need to convert the coordinates to the standard Cartesian coordinate system before applying the translation rule.
Q: Can I apply the translation rule to a point with complex coordinates?
A: Yes, you can apply the translation rule to a point with complex coordinates. However, you will need to add 2 to the real part of the coordinate and subtract 8 from the imaginary part.
Q: Can I apply the translation rule to a point with vector coordinates?
A: Yes, you can apply the translation rule to a point with vector coordinates. However, you will need to add 2 to the x-component of the vector and subtract 8 from the y-component.
Q: Can I apply the translation rule to a point with parametric coordinates?
A: Yes, you can apply the translation rule to a point with parametric coordinates. However, you will need to add 2 to the x-component of the parametric equation and subtract 8 from the y-component.
Q: Can I apply the translation rule to a point with polar coordinates?
A: Yes, you can apply the translation rule to a point with polar coordinates. However, you will need to add 2 to the radial distance and subtract 8 from the angle.
Q: Can I apply the translation rule to a point with spherical coordinates?
A: Yes, you can apply the translation rule to a point with spherical coordinates. However, you will need to add 2 to the radial distance and subtract 8 from the polar angle.
Q: Can I apply the translation rule to a point with cylindrical coordinates?
A: Yes, you can apply the translation rule to a point with cylindrical coordinates. However, you will need to add 2 to the radial distance and subtract 8 from the azimuthal angle.
Q: Can I apply the translation rule to a point with other types of coordinates?
A: Yes, you can apply the translation rule to a point with other types of coordinates. However, you will need to convert the coordinates to the standard Cartesian coordinate system before applying the translation rule.
Q: Can I apply the translation rule to a set of points?
A: Yes, you can apply the translation rule to a set of points. Simply apply the translation rule to each point in the set.
Q: Can I apply the translation rule to a function?
A: Yes, you can apply the translation rule to a function. However, you will need to apply the translation rule to each point in the domain of the function.
Q: Can I apply the translation rule to a curve?
A: Yes, you can apply the translation rule to a curve. However, you will need to apply the translation rule to each point on the curve.
Q: Can I apply the translation rule to a surface?
A: Yes, you can apply the translation rule to a surface. However, you will need to apply the translation rule to each point on the surface.
Q: Can I apply the translation rule to a 3D object?
A: Yes, you can apply the translation rule to a 3D object. However, you will need to apply the translation rule to each point on the object.
Q: Can I apply the translation rule to a 4D object?
A: Yes, you can apply the translation rule to a 4D object. However, you will need to apply the translation rule to each point on the object.
Q: Can I apply the translation rule to a higher-dimensional object?
A: Yes, you can apply the translation rule to a higher-dimensional object. However, you will need to apply the translation rule to each point on the object.
Q: Can I apply the translation rule to a fractal?
A: Yes, you can apply the translation rule to a fractal. However, you will need to apply the translation rule to each point on the fractal.
Q: Can I apply the translation rule to a geometric shape?
A: Yes, you can apply the translation rule to a geometric shape. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a polygon?
A: Yes, you can apply the translation rule to a polygon. However, you will need to apply the translation rule to each point on the polygon.
Q: Can I apply the translation rule to a polyhedron?
A: Yes, you can apply the translation rule to a polyhedron. However, you will need to apply the translation rule to each point on the polyhedron.
Q: Can I apply the translation rule to a complex shape?
A: Yes, you can apply the translation rule to a complex shape. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a shape with holes?
A: Yes, you can apply the translation rule to a shape with holes. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a shape with a hole in the middle?
A: Yes, you can apply the translation rule to a shape with a hole in the middle. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a shape with multiple holes?
A: Yes, you can apply the translation rule to a shape with multiple holes. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a shape with a hole in the corner?
A: Yes, you can apply the translation rule to a shape with a hole in the corner. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a shape with multiple holes in the corner?
A: Yes, you can apply the translation rule to a shape with multiple holes in the corner. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a shape with a hole in the middle and multiple holes in the corner?
A: Yes, you can apply the translation rule to a shape with a hole in the middle and multiple holes in the corner. However, you will need to apply the translation rule to each point on the shape.
Q: Can I apply the translation rule to a shape with multiple holes in the middle and multiple holes in the corner?
A: Yes, you can apply the translation rule to a shape with multiple holes