Perpendicular lines have opposite and reciprocal slopes, then two lines are perpendicular if the following expression is met:
[tex]m1=-\frac{1}{m2}[/tex]Where m1 is the slope of the first line and m2 is the slope of the second one, we can rewrite this expression to get:
[tex]\begin{gathered} m1\times m2=-\frac{1}{m2}\times m2 \\ m1\times m2=-1 \end{gathered}[/tex]Then, when we multiply the slopes of both lines the result must be -1, otherwise their slopes are not opposite and reciprocal.
The slope of a line m can be calculated by means of the following formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Where (x1, y1) and (x2, y2) are two points where the line passes through. For line BC we take the points (1, 2) and (4, 5), then we get:
[tex]mbc=\frac{5-2}{4-1}=\frac{3}{3}=1[/tex]For the line EF, we take the points (2, -1) and (5, -4) to get:
[tex]mef=\frac{-4-(-1)}{5-2}=\frac{-4+1}{3}=\frac{-3}{3}=-1[/tex]Now that we know the values of both slopes let's multiply them:
mbc×mef = 1×(-1) = -1
As you can see, we got -1, then the lines BC and EF have opposite reciprocal slopes.
