Exploring the Reflection of Points across the Line x -3

In coordinate geometry, reflections play a crucial role in understanding spatial transformations. We will discuss the reflection of the point (5, -2) across the line x -3, focusing on the principles and calculations involved in such transformations.

Understanding Reflection Across x -3

When reflecting a point across the vertical line x -3, the reflection retains the y-coordinate of the original point but changes the x-coordinate based on the distance from the line of reflection.

Distance Calculation and Reflection Formula

To find the reflection of the point (5, -2), the first step is to calculate the distance from the point to the line of reflection. The line x -3 is parallel to the y-axis and is 3 units to the left of the y-axis. The point (5, -2) is 8 units away from the line of reflection as shown below:

Distance |5 - (-3)| 8

Thus, the reflected point will be 8 units to the left of the line x -3. The abscissa (x-coordinate) of the reflected point is calculated as:

New x-coordinate -3 - 8 -11

Therefore, the coordinates of the reflected point are:

(-11, -2)

Construction of a Reflective Machine

To systematically generate the reflection points about the line x -3, we can use a machine or algorithm. This machine takes any point and generates its reflection by changing the abscissa and keeping the ordinate (y-coordinate) constant. Here is how it works:

Reflection Algorithm

Consider any point (x, y). The reflection of this point across the line x -3 is calculated by the transformation rule:

Reflected Point (-x - 6, y)

For the point (5, -2), we apply the transformation:

Reflected Point (-(5) - 6, -2) (-11, -2)

This confirms our earlier calculation that the point (5, -2) reflects to the point (-11, -2).

Line of Reflection and Perpendicular Distance

Before reflecting an object around a line, it is essential to identify the line of reflection, which is perpendicular to the distance between the object and the line. In this case, the line is x -3, and the distance from the point (5, -2) to this line is 8 units.

To find the reflected point, we measure this distance from the line of reflection, moving in the opposite direction:

Coordinate of reflected point: (-3 - 8, -2) (-11, -2)

Thus, the abscissa of the image point is -11, while the ordinate remains -2.

Conclusion

In summary, the reflection of the point (5, -2) across the line x -3 results in the point (-11, -2). This transformation involves only a change in the x-coordinate while keeping the y-coordinate constant. Understanding these principles is crucial for solving problems in geometry and spatial transformations.