Question

A video 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. Complete the explanation of why the red ant and the black ant are always the same distance from the anthill.

Answer 1

In order in which the boxes appear:

`\boxed{a√(2)}`

`\boxed{a√(2)}`

`\boxed{a√(2)}`

Explanation:

At any coordinate (x , y) the distance from the origin (0, 0) is computed by the distance (Pythagorean formula) as:

d =
`√(x^2+y^2)`

Since x = y = a for both ants, the distance is

`d = √(a^2+a^2)\\\\d = √(2a^2)\\\\d = √(a^2)\cdot √(2)\\\\d = a√(2)`

It does not matter whether both coordinates are positive or both negative since we are taking the squares of the coordinates and distance is always positive