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.

If you were to graph this situation out, you will see that the ants are traveling along the y = x graph.  It should be intuitive why they will always be the same distance from the anthill since they are moving at the same rate on the very same line, but in opposite directions.

Mathematically, we can use the distance formula.  Let (x,x) be the coordinate where the red ant is and and (-x,-x) be the coordinate where the black ant is, where "x" is some number.

So the red ant's distance from the origin is: sqrt[(x-0)^2 + (x-0)^2) = sqrt[2x^2]

= x*sqrt(2)

Similarly the black ant's distance from the origin is: sqrt[(-x-0)^2 + (-x-0)^2) = sqrt[2x^2]

= x*sqrt(2)

As you can see, they are the same distance from the anthill.