9.025 Parallel and perpendicular lines

Parallel lines

Two lines that are parallel have the same gradient. Since they have the same rise and run, they will always move in the same direction and thus never meet.

Thus two lines given by $y=m_1x+b$y=m1​x+b and $y=m_2x+c$y=m2​x+c will be parallel if and only if $m_1=m_2$m1​=m2​.

 

Perpendicular lines

Two lines are perpendicular if the product of their gradients is equal to $-1$−1. Another way to say this is that the gradient of one line is the negative reciprocal of the other.

That is, two lines given by $y=m_1x+b$y=m1​x+b and $y=m_2x+c$y=m2​x+c are perpendicular if and only if $m_1m_2=-1$m1​m2​=−1, or equivalently $m_1=-\frac{1}{m_2}$m1​=−1m2​​.

One way to visualise this relationship is to consider the rise and run of the lines as right triangles, with one triangle being a $90^\circ$90° rotation of the other. We can see that the lengths for rise and run are flipped, and the direction of one of them also changes. This is what gives us the negative reciprocal relationship.

 

Practice questions

Question 1

question 2