Question

explain why the dinstance formula is not needed to find the distance between two points that lie on a horinzontal or vertical line

Answer 1

The distance formula is

             D = square root of [ (difference in 'y')² + (difference in 'x')² ]

If the two points are on the same vertical line, the difference in 'x' is zero.
If they're on the same horizontal line, then their difference in 'y' is zero.

Either way, one of the terms in the distance formula is zero.  It goes away,
and the distance formula becomes ...

               D = square root of [ (difference in one coordinate)² ] .

That's just the square root of something squared, so it's only

               D  =  (difference in one coordinate)   

without going through all the squaring and rooting business.