Question

Area of parallelogram formed by lines y=mx,y=mx+1,y=nx,y=nx+1 (in form of m and n)?

Answer 1

the two pairs of parallel lines pass through y=0 and y=1
distance between these points =1,so b=1
This distance on the y-axis forms the base of a triangle which is half the area of the parallelogram.
The perpendicular height will be the distance between x=0 (y-axis) and the point of intersection of the lines y=mx and y=nx+1 (or the other pair)

y=mx . . . . . eqn 1
y=nx+1. . . . eqn 2
sub eqn 1 into eqn 2
mx = nx +1
mx - nx = 1
x(m-n) = 1
x = 1/ (m-n)

This is the x value and the distance the intersection is from the y-axis(perpendicular height)

Area = 2 * 1/2bh
Area = bh
Area = 1*1/(m-n)
Area = 1/ (m-n)

as m-n could be positive or negative depending on the values of m and n, you need to take the absolute value so the area will be positive

giving
Area = 1/ lm-nl