# Perpendicular Lines (Geometry)

Two lines are **perpendicular** or **orthogonal** if they meet at right angles.

For two perpendicular lines, all four angles formed by the two lines are equal to $90 ^ \circ$.

Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. In other words, the slopes of two perpendicular lines are negative reciprocals of each other. A vertical line has infinite slope and is perpendicular to only horizontal lines (which have slope 0).

Given the equations of two lines, this gives a method to test whether the two lines are perpendicular. This also gives us a way to find equations of lines that are perpendicular to another line. In the 2-dimensional plane, given a fixed line and any point on the line, there is exactly one line passing through this point and perpendicular to the original line.

In the following diagram, lines $l$ and $m$ are perpendicular lines. If $\angle a=34^\circ$, what is the measure of $\angle b$?

Since $l$ and $m$ are perpendicular lines, we have $\angle a + \angle b = 90^\circ$. Then $\angle a = 34^\circ$ gives

$\angle b = 90^\circ - \angle a = 90^\circ - 34^\circ = 56^\circ .\ _\square$

In the following diagram, lines $PS$, $QT$ and $RU$ meet at point $O$. Given that PS and UR are perpendicular lines, what is of $a + b$?

Angle $a = \angle QOR = \angle UOT$ (vertical angles). Angle $b = \angle TOS$. Hence

$a + b = \angle UOT + \angle TOS = \angle UOS = 90 ^ \circ.\ _\square$

In the following diagram, lines $l$ and $m$ are perpendicular lines. If $\angle a$ is 6 degrees less than $\angle b$, what is the measure of $\angle a$?

Since $\angle a$ is 6 degrees less than $\angle b$, we have $\angle a = \angle b - 6^\circ$. Since $l$ and $m$ are perpendicular lines, we have $\angle a + \angle b = 90^\circ$. Then substituting the first equation into the second equation gives

$\begin{aligned} \angle a + \angle b &= 90^\circ\\ (\angle b - 6^\circ) + \angle b &= 90^\circ\\ 2 \angle b &= 96^\circ\\ \angle b &= 48^\circ. \end{aligned}$

This gives $\angle a = \angle b - 6^\circ = 48^\circ - 6^\circ = 42^\circ. \ _\square$

Find the equation of the line perpendicular to

$y = -2x + 3$

and passing through the point $(-3, 9 )$.

The slope of the given line is -2. Then the slope of the perpendicular to the given line is

$m = - \frac{1}{-2} = \frac{1}{2}.$

Now, we have the slope of the perpendicular and a point on the line, so the equation of the perpendicular line is

$\begin{aligned} y - 9 & = \frac{1}{2} (x- (-3))\\ y - 9 &= \frac{x}{2} + \frac{3}{2}\\ y &= \frac{x}{2} +\frac{21}{2}. \ _\square \end{aligned}$

**Cite as:**Perpendicular Lines (Geometry).

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/perpendicular-lines/