reflection across x = -3 A graph is shown with multiple geometric points forming a shape. The points are labeled I, W, S, and P on a Cartesian plane.
The Correct Answer and Explanation is:
Here are the coordinates of the reflected quadrilateral:
I'(-1, -1)
P'(-2, -2)
S'(-4, 1)
W'(-4, 3)
Explanation
To find the reflection of the quadrilateral IWSP across the vertical line x = -3, we need to determine the new coordinates for each of its vertices. A reflection is a type of geometric transformation that flips a figure over a line, known as the line of reflection, to create a mirror image. This process is a rigid transformation, which means the resulting figure, I’W’S’P’, will have the same size and shape as the original.
First, we must identify the coordinates of the original vertices by reading the graph:
- I is located at (-5, -1).
- P is located at (-4, -2).
- S is located at (-2, 1).
- W is located at (-2, 3).
The line of reflection is the vertical line x = -3. For any reflection across a vertical line, the y-coordinate of each point remains unchanged. The new x-coordinate is found by measuring the horizontal distance from the original point to the line of reflection and then moving that same distance to the opposite side of the line.
Let’s apply this process to each vertex:
- Point I(-5, -1): The x-coordinate is -5. The horizontal distance from x = -5 to the line x = -3 is 2 units to the left. To reflect this point, we move 2 units to the right of the line x = -3. The new x-coordinate is -3 + 2 = -1. The y-coordinate stays the same. Therefore, the reflected point I’ is (-1, -1).
- Point P(-4, -2): The x-coordinate is -4, which is 1 unit to the left of the line x = -3. The reflection will be 1 unit to the right of the line. The new x-coordinate is -3 + 1 = -2. The y-coordinate remains -2. Thus, the reflected point P’ is (-2, -2).
- Point S(-2, 1): The x-coordinate is -2, which is 1 unit to the right of the line x = -3. The reflection will be 1 unit to the left of the line. The new x-coordinate is -3 – 1 = -4. The y-coordinate remains 1. So, the reflected point S’ is (-4, 1).
- Point W(-2, 3): The x-coordinate is -2, which is 1 unit to the right of the line x = -3. The reflection will be 1 unit to the left of the line. The new x-coordinate is -3 – 1 = -4. The y-coordinate remains 3. Consequently, the reflected point W’ is (-4, 3).
By connecting these new points, we form the reflected quadrilateral I’W’S’P’.
