Avideo game designer places an anthill at the origin of a coordinate plane. a red ant leaves the anthill and moves along a straight line to (1,1), while a black ant leaves the anthill and moves along a straight line to (-1,-1). next, the red ant moves to (2,2), while the black ant moves to (-2,-2). then the red ant moves to (3,3) while the black ant moves to (-3,-3), and so on. explain why the red ant and the black ant are always the same distance from the anthill.

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step-by-step explanation:

for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.