Suppose a coordinate grid is transposed over the blueprint for a house under construction, and two lines on the blueprint are perpendicular to each other. If one of the lines has the equation \begin{align*}y= \frac{1}{2} x + 1\end{align*}, and the other line passes through the point (3, 3), what is the equation of the second line?

### Equations of Perpendicular Lines

Lines can be parallel, **coincident** (overlap each other), or intersecting (crossing). Lines that intersect at \begin{align*}90^\circ\end{align*} angles have a special name: perpendicular lines. The slopes of perpendicular lines have a special property.

**Perpendicular lines** form a right angle. The product of their slopes is –1.

\begin{align*}m_1 \cdot m_2=-1\end{align*}

#### Let's verify that the following lines are perpendicular:

Line \begin{align*}a\end{align*}: passes through points (–2, –7) and (1, 5) Line \begin{align*}b\end{align*}: passes through points (4, 1) and (–8, 4)

Find the slopes of each line.

\begin{align*}Line \ a: \frac{5-(-7)}{1-(-2)}=\frac{12}{3}=\frac{4}{1} && Line \ b: \frac{4-1}{-8-4}=\frac{3}{-12}=\frac{-1}{4}\end{align*}

To verify that the lines are perpendicular, the product of their slopes must equal –1.

\begin{align*}\frac{4}{1} \times \frac{-1}{4}=-1\end{align*}

Because the product of their slopes is \begin{align*}-1\end{align*}, lines \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are perpendicular.

#### Now, let's determine whether the following two lines are parallel, perpendicular, or neither:

Line 1: \begin{align*}2x=y-10\end{align*}; Line 2: \begin{align*}y=-2x+5\end{align*}

Begin by finding the slopes of lines 1 and 2.

\begin{align*}2x+10& =y-10+10\\ 2x+10& =y\end{align*}

The slope of the first line is 2.

\begin{align*}y=-2x+5\end{align*}

The slope of the second line is –2.

These slopes are not identical, so these lines are not parallel.

To check if the lines are perpendicular, find the product of the slopes. \begin{align*}2 \times -2=-4\end{align*}. The product of the slopes is not –1, so the lines are not perpendicular.

Lines 1 and 2 are neither parallel nor perpendicular.

#### Writing Equations of Perpendicular Lines

Writing equations of perpendicular lines is slightly more difficult than writing parallel line equations. The reason is because you must find the slope of the perpendicular line before you can proceed with writing an equation.

#### Let's find the equation of the following line:

Find the equation of the line perpendicular to the line \begin{align*}y=-3x+5\end{align*} that passes through point (2, 6).

Begin by finding the slopes of the perpendicular line. Using the perpendicular line definition, \begin{align*}m_1 \cdot m_2=-1\end{align*}. The slope of the original line is –3. Substitute that for \begin{align*}m_1\end{align*}.

\begin{align*}-3 \cdot m_2=-1\end{align*}

Solve for \begin{align*}m_2\end{align*}, the slope of the perpendicular line.

\begin{align*}\frac{-3m_2}{-3}& =\frac{-1}{-3}\\ m_2& =\frac{1}{3}\end{align*}

The slope of the line perpendicular to \begin{align*}y=-3x+5\end{align*} is \begin{align*}\frac{1}{3}\end{align*}.

You now have the slope and a point. Use point-slope form to write its equation.

\begin{align*}y-6= \frac{1}{3}(x-2)\end{align*}

You can rewrite this in slope-intercept form: \begin{align*}y=\frac{1}{3} x-\frac{2}{3}+6\end{align*}.

\begin{align*}y=\frac{1}{3} x+\frac{16}{3}\end{align*}

### Examples

#### Example 1

Earlier, you were asked to suppose that when a coordinate grid is transposed over the blueprint of a house under construction, two lines on the blueprint are perpendicular. If one of the lines had the equation \begin{align*}y= \frac{1}{2} x + 1\end{align*} and the other line passes through (3,3), what would be the equation of the second line.

First, we need to determine what the slope of the other line is. Since the two lines are perpendicular, we can substitute the slope of the first line, \begin{align*}\frac{1}{2}\end{align*}, for \begin{align*}m_1\end{align*} in the perpendicular line definition, \begin{align*}m_1 \cdot m_2=-1\end{align*}.

\begin{align*}\frac{1}{2} \cdot m_2=-1\end{align*}

Solve for \begin{align*}m_2\end{align*}, the slope of the perpendicular line.

\begin{align*}(\frac{1}{2}m_2)\times2& =-1\times2\\ m_2& =-2\end{align*}

The slope of the line perpendicular to \begin{align*}y=\frac{1}{2}x+1\end{align*} is -2.

You now have the slope and a point. Use point-slope form to write its equation.

\begin{align*}y-3= -2(x-3)\end{align*}

You can rewrite this in slope-intercept form: \begin{align*}y=-2x+6+3\\ y = -2x + 9\end{align*}

The equation of the second line is \begin{align*}y = -2x + 9\end{align*}.

#### Example 2

Find the equation of the line perpendicular to the line \begin{align*}y=5\end{align*} and passing through (5, 4).

The line \begin{align*}y=5\end{align*} is a horizontal line with a slope of zero.

Lines that make a \begin{align*}90^\circ\end{align*} angle with a horizontal line are vertical lines.

Vertical lines are in the form \begin{align*}x=constant\end{align*}.

Since the vertical line must go through (5, 4), the equation is \begin{align*}x=5\end{align*}.

### Review

- Define perpendicular lines.
- What is true about the slopes of perpendicular lines?

In 3-7, determine the slope of a line perpendicular to the line given.

- \begin{align*}y=-5x+7\end{align*}
- \begin{align*}2x+8y=9\end{align*}
- \begin{align*}x=8\end{align*}
- \begin{align*}y=-4.75\end{align*}
- \begin{align*}y-2= \frac{1}{5}(x+3)\end{align*}

In 8–14, determine whether the lines are parallel, perpendicular, or neither.

- Line \begin{align*}a:\end{align*} passing through points (–1, 4) and (2, 6); Line \begin{align*}b:\end{align*} passing through points (2, –3) and (8, 1).
- Line \begin{align*}a:\end{align*} passing through points (4, –3) and (–8, 0); Line \begin{align*}b:\end{align*} passing through points (–1, –1) and (–2, 6).
- Line \begin{align*}a:\end{align*} passing through points (–3, 14) and (1, –2); Line \begin{align*}b:\end{align*} passing through points (0, –3) and (–2, 5).
- Line \begin{align*}a:\end{align*} passing through points (3, 3) and (–6, –3); Line \begin{align*}b:\end{align*} passing through points (2, –8) and (–6, 4).
- Line 1: \begin{align*}4y+x=8\end{align*}; Line 2: \begin{align*}12y+3x=1\end{align*}
- Line 1: \begin{align*}5y+3x+1\end{align*}; Line 2: \begin{align*}6y+10x=-3\end{align*}
- Line 1: \begin{align*}2y-3x+5=0\end{align*}; Line 2: \begin{align*}y+6x=-3\end{align*}

In 15-21, find the line perpendicular to it through the given point.

- \begin{align*}x+4y=12; (-3,-2)\end{align*}
- \begin{align*}y=\frac{1}{3}x+2; (-3,-1)\end{align*}
- \begin{align*}y=\frac{3}{5}x-4; (6,-2)\end{align*}
- \begin{align*}2x+y=5; (2,-2)\end{align*}
- \begin{align*}y=x-6; (-2,0)\end{align*}
- \begin{align*}5x-7=3y; (8,-2)\end{align*}
- \begin{align*}y=\frac{2}{3}x-1; (4,7)\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.8.