Point A(7,3) Translated To A'(16,-9) Finding The Translation Rule
In coordinate geometry, translations are fundamental transformations that shift a point or a shape from one location to another without changing its size or orientation. Understanding translations is crucial for various applications in mathematics, computer graphics, and physics. This article will delve into the concept of translations and provide a step-by-step guide to determine the translation rule when a point is moved from one position to another. We will specifically address the question: Point A(7,3) is translated to A'(16,-9). Which rule describes the translation?
What is a Translation?
A translation is a rigid transformation, meaning it preserves distances and angles. Imagine sliding a shape across a plane; that's essentially what a translation does. In the coordinate plane, a translation is defined by how much the x-coordinate and y-coordinate change. This change is constant for every point in the shape being translated. Mathematically, we represent a translation rule as follows:
(x, y) → (x + a, y + b)
Here,
- (x, y) represents the original coordinates of a point.
- (x + a, y + b) represents the coordinates of the translated point.
- 'a' is the horizontal shift (positive for right, negative for left).
- 'b' is the vertical shift (positive for up, negative for down).
Our goal is to find the values of 'a' and 'b' that describe the translation from point A(7,3) to A'(16,-9).
Step-by-Step Solution to Determine the Translation Rule
To determine the translation rule, we need to find the horizontal and vertical shifts that map point A to point A'. Let's break this down step by step:
1. Understand the Given Information
We are given two points:
- Point A (Original Point): A(7, 3)
- Point A' (Translated Point): A'(16, -9)
We need to find the values 'a' and 'b' in the translation rule (x, y) → (x + a, y + b).
2. Calculate the Horizontal Shift (a)
The horizontal shift 'a' is the difference between the x-coordinate of the translated point (A') and the x-coordinate of the original point (A). We can calculate it as follows:
a = x' - x
Where:
- x' is the x-coordinate of A' (16).
- x is the x-coordinate of A (7).
Substituting the values, we get:
a = 16 - 7 = 9
This means the point has been shifted 9 units horizontally. Since 'a' is positive, the shift is to the right.
3. Calculate the Vertical Shift (b)
Similarly, the vertical shift 'b' is the difference between the y-coordinate of the translated point (A') and the y-coordinate of the original point (A). We calculate it as follows:
b = y' - y
Where:
- y' is the y-coordinate of A' (-9).
- y is the y-coordinate of A (3).
Substituting the values, we get:
b = -9 - 3 = -12
This means the point has been shifted 12 units vertically. Since 'b' is negative, the shift is downwards.
4. Formulate the Translation Rule
Now that we have the horizontal shift (a = 9) and the vertical shift (b = -12), we can formulate the translation rule. Recall the general form:
(x, y) → (x + a, y + b)
Substituting the values of 'a' and 'b', we get:
(x, y) → (x + 9, y - 12)
This rule describes the translation that maps point A(7, 3) to point A'(16, -9).
Analyzing the Answer Choices
Now, let's compare our result with the given answer choices:
A. (x, y) → (x - 9, y - 12) B. (x, y) → (x - 9, y + 12) C. (x, y) → (x + 9, y + 12) D. (x, y) → (x + 9, y - 12)
Our calculated translation rule (x, y) → (x + 9, y - 12) matches option D. Therefore, the correct answer is:
D. (x, y) → (x + 9, y - 12)
Importance of Understanding Translations
Translations are not just abstract mathematical concepts; they have practical applications in various fields:
- Computer Graphics: In computer graphics, translations are used to move objects around the screen. For example, when you move a character in a video game, you are essentially applying translation transformations.
- Physics: In physics, translations are used to describe the movement of objects in space. The displacement of an object is a translation vector.
- Engineering: Engineers use translations in CAD (Computer-Aided Design) software to position components in a design.
- Mapping and Navigation: Translations are fundamental in mapping and navigation systems, helping to represent movement across geographical spaces.
Understanding translations provides a foundation for more advanced geometric transformations such as rotations and reflections, which are essential in fields ranging from robotics to animation.
Common Mistakes and How to Avoid Them
When working with translations, students often make a few common mistakes. Understanding these pitfalls can help you avoid them:
1. Confusing the Order of Subtraction
One common mistake is subtracting the coordinates in the wrong order when calculating the horizontal and vertical shifts. Remember, you need to subtract the original coordinates from the translated coordinates:
- a = x' - x (translated x - original x)
- b = y' - y (translated y - original y)
2. Incorrectly Interpreting Signs
Another common mistake is misinterpreting the signs of the shifts. A positive horizontal shift means the point moves to the right, while a negative shift means it moves to the left. Similarly, a positive vertical shift means the point moves up, and a negative shift means it moves down. Be careful with the signs, as they indicate the direction of the translation.
3. Applying the Shift in Reverse
Sometimes, students mistakenly add the shifts to the translated point instead of the original point. Remember, the translation rule is applied to the original point to get the translated point. Double-check that you are applying the shifts in the correct direction.
4. Not Understanding the Concept of Rigid Transformations
It's important to remember that translations are rigid transformations. This means that the size and shape of the figure do not change during the translation. If your calculations lead to a change in size or shape, there's likely an error in your approach.
By keeping these common mistakes in mind and carefully checking your work, you can confidently solve translation problems.
Practice Problems
To solidify your understanding of translations, let's work through a few practice problems:
Practice Problem 1
Point B(-2, 5) is translated to B'(3, 1). Find the translation rule.
Solution:
- Calculate the horizontal shift: a = 3 - (-2) = 5
- Calculate the vertical shift: b = 1 - 5 = -4
- The translation rule is (x, y) → (x + 5, y - 4)
Practice Problem 2
A triangle with vertices C(1, 2), D(4, 2), and E(4, 5) is translated using the rule (x, y) → (x - 3, y + 1). Find the coordinates of the translated vertices C', D', and E'.
Solution:
- C'(1 - 3, 2 + 1) = C'(-2, 3)
- D'(4 - 3, 2 + 1) = D'(1, 3)
- E'(4 - 3, 5 + 1) = E'(1, 6)
Practice Problem 3
If the point F(6, -4) is translated using the rule (x, y) → (x + a, y + b) and the image is F'(2, -1), find the values of 'a' and 'b'.
Solution:
- a = 2 - 6 = -4
- b = -1 - (-4) = 3
- Therefore, the translation rule is (x, y) → (x - 4, y + 3), and a = -4, b = 3.
By working through these practice problems, you can reinforce your understanding of translation rules and improve your problem-solving skills.
Conclusion
In this article, we explored the concept of translations in coordinate geometry and provided a detailed, step-by-step guide to determine the translation rule when a point is moved from one position to another. We addressed the specific question: Point A(7,3) is translated to A'(16,-9). Which rule describes the translation? We found that the correct answer is (x, y) → (x + 9, y - 12). Understanding translations is essential for various applications in mathematics, computer graphics, and physics. By mastering this fundamental transformation, you build a strong foundation for more advanced concepts in geometry and related fields. Remember to avoid common mistakes and practice regularly to reinforce your skills. Translations are a key building block in the world of geometric transformations, and mastering them opens the door to understanding more complex spatial relationships and applications.
