The triangle with vertices A(1,3) B(-2,3) and C(3,5) is reflected over the line y=x. a) Graph the triangle and its image A,B,C, b)What is the image mapping over the line y=x? (x,y) -> ( ? , ?)

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We have a triangle with vertices A at (1,3), B at (-2,3), and C at (3,5). We need to reflect this triangle over the line y equals x. The reflection line y equals x is shown in purple. When we reflect over this line, the coordinates swap: x becomes y and y becomes x. The rule for reflecting over the line y equals x is simple: swap the coordinates. Point (x,y) becomes point (y,x). Let's see this in action. Point A at (1,3) becomes A prime at (3,1). Point B at (-2,3) becomes B prime at (3,-2). Point C at (3,5) becomes C prime at (5,3). Notice how each point and its reflection are connected by a line perpendicular to y equals x. Reflection over y equals x has several important properties. The distance from each point to the line y equals x is preserved. The connecting lines between original and reflected points are perpendicular to the reflection line. The line y equals x acts as the perpendicular bisector of each connecting segment, shown by the green midpoints. The shape and size of the triangle are preserved, but the orientation is reversed. Let's go through the reflection process step by step. First, we identify the original vertices: A at (1,3), B at (-2,3), and C at (3,5). Next, we apply the reflection rule (x,y) becomes (y,x) to get the reflected vertices. Then we plot these reflected points: A prime at (3,1), B prime at (3,-2), and C prime at (5,3). Finally, we connect these points to form the reflected triangle. The general mapping rule is (x,y) maps to (y,x). To summarize our complete solution: Part a asks us to graph the triangle and its image. The original triangle has vertices A at (1,3), B at (-2,3), and C at (3,5). After reflection over y equals x, the image triangle has vertices A prime at (3,1), B prime at (3,-2), and C prime at (5,3). Part b asks for the image mapping rule, which is (x,y) maps to (y,x). This means we simply swap the x and y coordinates to find the reflected point.