Verified Answer :
- Identify the reflection rule for the line \(y = x\), which swaps the \(x\) and \(y\) coordinates of any point.
- Apply the mapping \((x, y) \to (y, x)\) to each of the given vertices: \(A(3, 2)\), \(B(-4, 1)\), and \(C(-3, -2)\).
- Obtain the image coordinates: \(A'(2, 3)\), \(B'(1, -4)\), and \(C'(-2, -3)\).
- The final coordinates are \(\boxed{A'(2, 3), B'(1, -4), C'(-2, -3)}\).
Explanation
- Define the transformation rule.
To find the coordinates of the vertices of triangle \(ABC\) after a reflection across the line \(y = x\), we use the standard transformation rule for reflection in this line. The rule states that a point \((x, y)\) is mapped to its image \((x', y')\) by swapping its coordinates: \((x', y') = (y, x)\). This transformation essentially mirrors any point across the line where the x-coordinate and y-coordinate are equal.
- Apply the transformation to the vertices.
We apply the rule \((x, y) \to (y, x)\) to each vertex of the triangle:
- For vertex \(A(3, 2)\):
The x-coordinate is 3 and the y-coordinate is 2. Swapping these gives the new coordinates \((2, 3)\). Thus, \(A' = (2, 3)\).
- For vertex \(B(-4, 1)\):
The x-coordinate is -4 and the y-coordinate is 1. Swapping these gives the new coordinates \((1, -4)\). Thus, \(B' = (1, -4)\).
- For vertex \(C(-3, -2)\):
The x-coordinate is -3 and the y-coordinate is -2. Swapping these gives the new coordinates \((-2, -3)\). Thus, \(C' = (-2, -3)\).
- State the final coordinates.
By applying the reflection rule, we have determined the new coordinates for each vertex. The vertices of the reflected triangle \(A'B'C'\) are \(A'(2, 3)\), \(B'(1, -4)\), and \(C'(-2, -3)\). This concludes the process of reflecting the triangle across the line \(y = x\). The final coordinates are \(\boxed{A'(2, 3), B'(1, -4), C'(-2, -3)}\).
Examples
Reflections are fundamental in geometry and daily life. Think of a mirror placed along the line \(y = x\); if you were to place objects at the positions of vertices \(A\), \(B\), and \(C\), their mirror images would appear exactly at the points \(A'\), \(B'\), and \(C'\). This principle is widely used in computer graphics to create symmetric designs, in architecture to analyze structural balance, and in art to generate intricate, mirrored patterns that are visually pleasing. Understanding coordinate transformations allows you to manipulate and analyze these shapes precisely!