If a(x1, y1), b(x2, y2), c(x3, y3), and d(x4, y4) form two line segments, and , which of these conditions needs to be met to prove that. ?

You are asking for the condition needed for the 
two segments to be perpendicular to each other. 

If the slope of one segment is m, then the slope of a perpendicular segment would be -1/m 

this means that m2 = -1/m and so m2 x m = -1 

If you look carefully at your choices, the 3rd answer involves the two slope and has the -1. It's 
the correct answer.

The condition that seems to fit the bill in regards to, two line segments to be actually perpendicular to one another is "C" or the third choice (x3, y3). Now the obvious question is to how one gets to this conclusion? Now if one considers one segment to be equal to "m", then the perpendicular segment to it would be "-1/m". This also means
m^2 = -1/m
m^2 * m = -1
This actually proves that "C" is the correct answer.