Question

Determine if lines passing through the points are Parallel,perpendicular,or neither line 1:(-2,2)and (2,-4) line 2:(3,6) and (5,3)

Answer 1

You can tell if two lines are parallel, perpendicular, or neither by looking at their slopes $m_1$ and $m_2$:

• If $m_1=m_2$, i.e. if the two lines have the same slope, the lines are parallel
• If $m_1\cdot m_2=-1$, the lines are perpendicular
• In all other cases, the lines are not parallel nor perpendicular.

Given two points $A = (x_A,y_A),\ B = (x_B,y_B)$ of a line, the slope is defined as the ratio between the y and x variation:

$m = \dfrac{\Delta y}{\Delta x} = \dfrac{y_B-y_A}{x_B-x_A}$

So in this case, we have

$m_1 = \dfrac{2-(-4)}{-2-2} = \dfrac{6}{-4} = -\dfrac{3}{2}$

$m_2 = \dfrac{3-6}{5-3} = \dfrac{-3}{2} = -\dfrac{3}{2}$

Since the two slopes are the same, the two lines are parallel.