In which scenario would the distance formula not yield a positive value?

The distance formula calculates the distance between two points in a coordinate system using the formula ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ). This formula will yield a non-negative value for any two points because it involves squaring the differences in their coordinates, which eliminates any negative values.

When points are identical, their coordinates are exactly the same, meaning both ( x_1 ) and ( x_2 ) are equal, as are ( y_1 ) and ( y_2 ). Therefore, the result of the distance formula would be ( d = \sqrt{(0)^2 + (0)^2} = \sqrt{0} = 0 ). Here, the distance calculated is zero, which is neither positive nor negative. Thus, this scenario is indeed the only case where the distance formula fails to produce a positive value, as it accurately reflects the fact that the points occupy the same position in space.

The other scenarios outlined in the choices will always ensure that the distance calculated is either zero or positive. For instance, points that lie on a straight line, are plotted on a graph, or are in opposite