Finding The Y-coordinate Of P' After Translation Of Triangle PQR

In the fascinating realm of coordinate geometry, geometric transformations play a pivotal role in altering the position and orientation of shapes in a coordinate plane. Among these transformations, translation holds a special place as a fundamental operation that shifts a figure without altering its size or shape. In this comprehensive exploration, we delve into the intricacies of translation, specifically focusing on its application to a triangle in the coordinate plane. We will embark on a journey to understand how translation affects the coordinates of a triangle's vertices and, more importantly, how to determine the new coordinates of a translated point. Let's consider a triangle PQR with vertices located at specific coordinates: P(-2, 6), Q(-8, 4), and R(1, -2). Imagine this triangle undergoing a transformation, a shift across the coordinate plane, guided by a specific rule. This rule dictates how each point of the triangle is repositioned, a systematic movement that defines the translation. In our case, the translation rule is defined as (x, y) → (x - 2, y - 16). This rule signifies that every point (x, y) in the original triangle will be shifted 2 units to the left (due to the x - 2 component) and 16 units downward (due to the y - 16 component). The question at hand is to determine the y-coordinate of the transformed point P', which is the image of point P after the translation. This seemingly simple question opens the door to a deeper understanding of translation and its effects on coordinate points. By carefully applying the translation rule, we can unravel the mystery of P's y-coordinate, gaining valuable insights into the world of geometric transformations.

Understanding Translation in Coordinate Geometry

In the context of coordinate geometry, a translation is a rigid transformation that shifts a geometric figure from one location to another without changing its size, shape, or orientation. This shift is defined by a translation vector, which specifies the magnitude and direction of the movement. Mathematically, a translation can be represented by a rule that maps each point (x, y) in the original figure to a new point (x', y') in the translated figure. This rule typically takes the form (x, y) → (x + a, y + b), where 'a' and 'b' are constants that determine the horizontal and vertical shifts, respectively. A positive value of 'a' indicates a shift to the right, while a negative value indicates a shift to the left. Similarly, a positive value of 'b' indicates an upward shift, while a negative value indicates a downward shift. In our specific problem, the translation rule is given as (x, y) → (x - 2, y - 16). This rule tells us that each point in the triangle PQR will be shifted 2 units to the left (since a = -2) and 16 units downward (since b = -16). To find the coordinates of the translated triangle P'Q'R', we need to apply this rule to each of the original vertices P, Q, and R. For example, to find the coordinates of P', we substitute the coordinates of P (-2, 6) into the translation rule: P' ((-2) - 2, 6 - 16) = P' (-4, -10). Similarly, we can find the coordinates of Q' and R' by applying the same rule to the coordinates of Q (-8, 4) and R (1, -2). The beauty of translation lies in its simplicity and predictability. It preserves the fundamental properties of the geometric figure, such as side lengths, angles, and area. This makes translation a powerful tool in various geometric applications, including computer graphics, robotics, and engineering.

Applying the Translation Rule to Point P

To determine the y-coordinate of P', the translated image of point P, we must meticulously apply the given translation rule (x, y) → (x - 2, y - 16). This rule serves as our guiding principle, dictating how each coordinate of point P will be transformed. Recall that point P has coordinates (-2, 6). This means that the x-coordinate of P is -2, and the y-coordinate of P is 6. Now, let's apply the translation rule to these coordinates. According to the rule, the x-coordinate of P' will be obtained by subtracting 2 from the x-coordinate of P. Therefore, the x-coordinate of P' is -2 - 2 = -4. Similarly, the y-coordinate of P' will be obtained by subtracting 16 from the y-coordinate of P. Therefore, the y-coordinate of P' is 6 - 16 = -10. Thus, the coordinates of P' are (-4, -10). The question specifically asks for the y-coordinate of P', which we have now determined to be -10. This value represents the vertical position of the translated point P' in the coordinate plane. It signifies that P' is located 10 units below the x-axis. By carefully applying the translation rule, we have successfully pinpointed the y-coordinate of P', demonstrating the power of coordinate geometry in describing and predicting geometric transformations. This process highlights the importance of understanding translation rules and their impact on the coordinates of points.

Detailed Solution

Let's embark on a step-by-step journey to solve this problem with clarity and precision. Our objective is to find the y-coordinate of P', the image of point P after the translation. We are given the coordinates of point P as (-2, 6) and the translation rule as (x, y) → (x - 2, y - 16). To find the coordinates of P', we must apply this rule to the coordinates of P. This involves performing the operations specified in the rule, which are subtracting 2 from the x-coordinate and subtracting 16 from the y-coordinate. 1. Identify the coordinates of point P: As stated earlier, point P has coordinates (-2, 6). This means x = -2 and y = 6. 2. Apply the translation rule to the x-coordinate: The translation rule dictates that the x-coordinate of P' is obtained by subtracting 2 from the x-coordinate of P. Therefore, the x-coordinate of P' is -2 - 2 = -4. 3. Apply the translation rule to the y-coordinate: Similarly, the y-coordinate of P' is obtained by subtracting 16 from the y-coordinate of P. Therefore, the y-coordinate of P' is 6 - 16 = -10. 4. State the coordinates of P': Combining the results from steps 2 and 3, we find that the coordinates of P' are (-4, -10). 5. Identify the y-coordinate of P': The question specifically asks for the y-coordinate of P', which we have determined to be -10. Therefore, the answer is -10. This detailed solution provides a clear and concise pathway to the answer, highlighting each step involved in applying the translation rule and arriving at the final result. By breaking down the problem into smaller, manageable steps, we can ensure accuracy and understanding. The y-coordinate of P' is a crucial piece of information that helps us locate the transformed point in the coordinate plane.

The Answer and Its Significance

After meticulously applying the translation rule to point P, we have definitively determined that the y-coordinate of P' is -10. This seemingly simple numerical value holds significant meaning in the context of geometric transformations. The y-coordinate, as we know, represents the vertical position of a point in the coordinate plane. A negative y-coordinate indicates that the point is located below the x-axis. In this case, the y-coordinate of -10 tells us that P' is situated 10 units below the x-axis. This information, combined with the x-coordinate of P' (-4), allows us to precisely pinpoint the location of P' in the coordinate plane. P' is located at the point (-4, -10), which is 4 units to the left of the y-axis and 10 units below the x-axis. The fact that the y-coordinate changed from 6 to -10 after the translation is a direct consequence of the vertical shift component in the translation rule. The rule (x, y) → (x - 2, y - 16) specifies a downward shift of 16 units. This downward shift is reflected in the change in the y-coordinate of P, which decreased by 16 units (from 6 to -10). Understanding the significance of the y-coordinate of P' allows us to visualize the effect of the translation on point P. It helps us grasp how the translation has repositioned the point in the coordinate plane, shifting it both horizontally and vertically. The y-coordinate is not just a number; it is a vital piece of information that provides insight into the geometric transformation that has occurred.

Why Other Options are Incorrect

In the realm of multiple-choice questions, understanding why the incorrect options are wrong is just as crucial as knowing why the correct option is right. Let's dissect the other options provided in this problem and understand why they do not represent the correct y-coordinate of P'. Option A: -10 This is the correct answer, as we have meticulously demonstrated through our step-by-step solution. Option B: -16 This option represents the vertical shift component in the translation rule, but it does not account for the original y-coordinate of point P. The y-coordinate of P' is not simply the vertical shift; it is the result of applying the shift to the original y-coordinate. Therefore, -16 is incorrect. Option C: -18 This option is likely the result of incorrectly adding the vertical shift component to the original y-coordinate instead of subtracting it. The translation rule specifies a subtraction of 16 from the y-coordinate, not an addition. Therefore, -18 is incorrect. Option D: -12 This option may be the result of a calculation error or a misunderstanding of the translation rule. It does not align with the correct application of the rule to the coordinates of point P. Therefore, -12 is incorrect. By analyzing these incorrect options, we gain a deeper appreciation for the importance of carefully applying the translation rule and avoiding common pitfalls. Each incorrect option represents a potential error in reasoning or calculation, highlighting the need for a thorough and systematic approach to problem-solving. Understanding why other options are incorrect reinforces our understanding of the correct solution and solidifies our grasp of the underlying concepts.

Conclusion

In conclusion, the y-coordinate of P', the image of point P after the translation, is -10. This result is obtained by carefully applying the translation rule (x, y) → (x - 2, y - 16) to the coordinates of point P (-2, 6). The translation rule dictates a horizontal shift of 2 units to the left and a vertical shift of 16 units downward. Applying this rule to the y-coordinate of P, we subtract 16 from 6, resulting in -10. This value represents the vertical position of P' in the coordinate plane, indicating that P' is located 10 units below the x-axis. Understanding the concept of translation and its effect on coordinate points is fundamental in coordinate geometry. Translation is a rigid transformation that shifts a figure without altering its size, shape, or orientation. The translation rule provides a systematic way to determine the new coordinates of a point after the translation. By carefully applying the rule, we can accurately predict the position of the translated point. The y-coordinate of P' is a crucial piece of information that allows us to visualize the effect of the translation on point P. It helps us understand how the translation has repositioned the point in the coordinate plane, shifting it both horizontally and vertically. This problem serves as a valuable exercise in applying the principles of coordinate geometry to solve a specific problem. It reinforces our understanding of translation, coordinate points, and the importance of careful calculation and reasoning. By mastering these concepts, we can confidently tackle more complex geometric problems and explore the fascinating world of geometric transformations.