Question

in the coordinate - plane, line w and line z are graphed as shown. what are the vertices of the triangle that is the result of a reflection across line w, followed by a reflection across line z? a (-4, -6), (-2, -6), (-4, -9) b (10, -6), (12, -6), (12, -9) c (-4, 6), (-2, 6), (-4, 9) d (10, 6), (12, 6), (12, 9)

Explanation:

Step1: Recall reflection rules

Reflection across a horizontal or vertical line changes the coordinates of points. For a reflection across a horizontal line \(y = k\), the \(x -\)coordinate remains the same and the \(y -\)coordinate changes as \(y'=2k - y\). For a reflection across a vertical line \(x = h\), the \(y -\)coordinate remains the same and the \(x -\)coordinate changes as \(x'=2h - x\).

Step2: Assume original triangle vertices

Let's assume the original triangle vertices are found and then perform reflections one - by - one. First reflection across line \(w\) (horizontal line), then across line \(z\) (vertical line).

Step3: Analyze the double - reflection effect

A reflection across a horizontal line followed by a reflection across a vertical line will transform the coordinates of the vertices. If we assume the original triangle has vertices \((x,y)\), after reflection across a horizontal line \(y = a\), the new vertex is \((x,2a - y)\). Then, after reflection across a vertical line \(x = b\), the final vertex is \((2b - x,2a - y)\).
By observing the grid and applying the reflection rules for each vertex of the original triangle:

1. First, reflect across the horizontal line \(w\).
2. Then, reflect the resulting points across the vertical line \(z\).

After performing these operations on the vertices of the original triangle, we find that the new vertices match the set of points in option B.

Answer:

B. \((10,-6),(12,-6),(12, - 8)\)