To reflect a point or shape over the line y = x, you simply swap the x and y coordinates of each point.
The line y = x is a diagonal line that passes through the origin (0,0) and has equal x and y coordinates for every point on it (e.g., (1,1), (-5,-5)). When reflecting an object across this line, its image appears on the opposite side, maintaining the same distance from the line of reflection.
The Fundamental Rule: Swapping Coordinates
The core principle for reflecting over the line y = x is to interchange the values of the x-coordinate and the y-coordinate. If your original point is (x, y), its reflected image will be (y, x).
In essence:
- The original x-coordinate becomes the new y-coordinate.
- The original y-coordinate becomes the new x-coordinate.
Step-by-Step Reflection Process
Follow these easy steps to reflect any point or figure over the line y = x:
- Identify Point Coordinates: Determine the (x, y) coordinates of the point you want to reflect. For shapes, identify the coordinates of all its vertices.
- Swap the Coordinates: Take the (x, y) coordinates and switch their positions. The new coordinates will be (y, x).
- Plot the Reflected Point: Plot these new (y, x) coordinates to locate the reflected image of your original point.
- Connect Vertices (for Shapes): If you are reflecting a shape, repeat this process for all its vertices. Then, connect these newly plotted reflected vertices in the same order as the original shape to form its image.
Practical Examples
Let's look at a few examples to solidify this concept:
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Example 1: Reflecting a point with a negative x-coordinate
- Original point: P = (-3, 1)
- To find the reflected point P', we swap the x and y values. The x-coordinate (-3) moves to the y-position, and the y-coordinate (1) moves to the x-position.
- Reflected point P' = (1, -3)
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Example 2: Reflecting a point with positive coordinates
- Original point: Q = (5, 2)
- Swap the x and y values:
- Reflected point Q' = (2, 5)
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Example 3: Reflecting a point on an axis
- Original point: R = (0, 4)
- Swap the x and y values:
- Reflected point R' = (4, 0)
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Example 4: Reflecting a point on the line y = x
- Original point: S = (3, 3)
- Swap the x and y values:
- Reflected point S' = (3, 3). Points that lie directly on the line of reflection remain unchanged.
Summary of Coordinate Transformations
The table below summarizes how different points transform when reflected over y = x:
| Original Point (x, y) | Reflected Point (y, x) |
|---|---|
| (A, B) | (B, A) |
| (-3, 1) | (1, -3) |
| (5, 2) | (2, 5) |
| (0, 4) | (4, 0) |
| (-6, -9) | (-9, -6) |
For a deeper understanding of transformations in geometry, including various types of reflections, you can explore resources such as Khan Academy's section on Reflections.
