What if you were given two parallel lines? How could you find how far apart these two lines are? After completing this Concept, you'll be able to find the distance between two vertical lines, two horizontal lines, and two non-vertical, non-horizontal parallel lines using the perpendicular slope.

### Watch This

CK-12 Finding The Distance Between Parallel Lines

James Sousa: Determining the Distance Between Two Parallel Lines

### Guidance

All vertical lines are in the form \begin{align*}x = a\end{align*}

In general, the shortest distance between two parallel lines is the length of a perpendicular segment between them. There are infinitely many perpendicular segments between two parallel lines, but they will all be the same length.

Remember that distances are always positive!

#### Example A

Find the distance between \begin{align*}x = 3\end{align*}

The two lines are 3 – (-5) units apart, or 8 units apart.

#### Example B

Find the distance between \begin{align*}y = 5\end{align*}

The two lines are 5 – (-8) units apart, or 13 units apart.

#### Example C

Find the distance between \begin{align*}y = x+6\end{align*}

Step 1: Find the perpendicular slope.

\begin{align*}m = 1\end{align*}

Step 2: Find the \begin{align*}y-\end{align*}

Step 3: Use the slope and count down 1 and to the right 1 until you hit \begin{align*}y=x-2\end{align*}

**Always rise/run the same amount for \begin{align*}m = 1\end{align*} m=1 or -1.**

Step 4: Use these two points in the distance formula to determine how far apart the lines are.

\begin{align*}d & = \sqrt{(0-4)^2 + (6-2)^2}\\ & = \sqrt{(-4)^2 + (4)^2}\\ & = \sqrt{16+16}\\ & = \sqrt{32} = 5.66 \ units\end{align*}

CK-12 Finding The Distance Between Parallel Lines

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### Guided Practice

1. Find the distance between \begin{align*}y=-x-1\end{align*}

2. Find the distance between \begin{align*}y = 2\end{align*}

3. Find the distance between \begin{align*}x = -5\end{align*}

**Answers:**

1. Step 1: Find the perpendicular slope.

\begin{align*}m = -1\end{align*}

Step 2: Find the \begin{align*}y-\end{align*}

Step 3: Use the slope and count down 1 and to the *left* 1 until you hit \begin{align*}y=x-3\end{align*}

Step 4: Use these two points in the distance formula to determine how far apart the lines are.

\begin{align*}d & =\sqrt{(0-(-1))^2 + (-1-(-2))^2}\\ & = \sqrt{(1)^2+(1)^2}\\ & = \sqrt{1+1}\\ & = \sqrt{2} = 1.41 \ units\end{align*}

2. The two lines are 2 – (-4) units apart, or 6 units apart.

3. The two lines are -5 – (-10) units apart, or 5 units apart.

### Explore More

Use each graph below to determine how far apart each pair of parallel lines is.

Determine the shortest distance between the each pair of parallel lines. Round your answer to the nearest hundredth.

- \begin{align*}x = 5, x = 1\end{align*}
- \begin{align*}y = -6, y = 4\end{align*}
- \begin{align*}y = 3, y = 15\end{align*}
- \begin{align*}x = -10, x = -1\end{align*}
- \begin{align*}x = 8, x = 0\end{align*}
- \begin{align*}y = 7, y = -12\end{align*}

Find the distance between the given parallel lines.

- \begin{align*}y=x-3, \ y=x+11\end{align*}
- \begin{align*}y=-x+4, \ y=-x\end{align*}
- \begin{align*}y=-x-5, \ y = -x+1\end{align*}
- \begin{align*}y = x+12 , \ y=x-6\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 3.11.