Aline segment has endpoints (-5, 10) and (a, b) . if the midpoint of the segment is (13, -2), what is the absolute difference between a and b ?

If the line segment point is D(-5, 10) and E(a,b) and the  midpoint of the segment is F(13, -2) that mean 
DE= 2*DF

You can directly find the distance of AC 
Xdf= Xf-Xd= 13 - (-5)= 18
Ydf= Yf - Yd= -2 - 10= -12

Then add the distance of AB( which is 2*AC) to point D
Xe= a = Xd + 2*Xdf 
a= -5 +2*18= 31

Ye= b = Yd + 2Yf
b= 10+ 2*-12= -14

absolute difference between a and b:
|b-a|= |-14-31|= 45