Question

triangle gmh, line w, and line x are graphed on the coordinate plane below. what are the vertices of the triangle that is the result of a reflection across line w, followed by a reflection across line x? 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 vertical or horizontal line changes the coordinates of points. Let's assume we first find the general rules for reflection across line \(w\) and then across line \(z\).

Step2: Analyze reflection across line \(w\)

When reflecting a point \((x,y)\) across a vertical or horizontal line, we use the distance - based rule. But we can also note that two reflections across two lines can be equivalent to a translation in some cases. If we assume line \(w\) and line \(z\) are parallel or perpendicular (from the context of coordinate - plane reflections), we can analyze the net transformation.

Step3: Consider the net transformation

Two reflections across two lines is equivalent to a rotation or translation. For a reflection across two perpendicular lines, it is equivalent to a \(180^{\circ}\) rotation about the intersection point of the two lines. Let's assume the original triangle has vertices and we consider the sign - change of coordinates due to the double - reflection. If we assume the original triangle has vertices \((x,y)\), after reflection across one line and then another perpendicular line, the new coordinates \((x',y')\) will have the signs of both \(x\) and \(y\) changed compared to the original in a \(180^{\circ}\) - like transformation.
Let's assume the original triangle has vertices in the first - quadrant like appearance. After double - reflection, the vertices will be in the third - quadrant (if the original was in the first) with negative \(x\) and \(y\) values.

Answer:

A. \((-4,-6),(-2,-6),(-4, - 9)\)