# Thread: Find exact perpendicular distance between two parallel lines

1. ## Find exact perpendicular distance between two parallel lines

Find the exact perpendiuclar distance between two parallel lines.
a)y=3x-2 and y=3x+3

2. ## Re: Find exact perpendicular distance between two parallel lines

Originally Posted by roger1505
Find the exact perpendiuclar distance between two parallel lines.
a)y=3x-2 and y=3x+3
There is a formula for the distance from a point to a line. FIND IT!
Then find a point on one of the parallel line, use that formula to find the distance from that point to the other line.

3. ## Re: Find exact perpendicular distance between two parallel lines

can you please do it. mainly because its a parallel, and i dont know how to sub it in

4. ## Re: Find exact perpendicular distance between two parallel lines

Your teacher did not show how...or you missed classes?

5. ## Re: Find exact perpendicular distance between two parallel lines

i wasnt in class when they taught this, but does anyone know?

6. ## Re: Find exact perpendicular distance between two parallel lines

You can easily use the "formula" for the distance between a point and a line. For example, you have $\displaystyle y = 3x-2$ so you know that the point (0,-2) is on the line. So use the formula to find the distance between (0,-2) and the line $\displaystyle y = 3x-2$.

Or, you can skip the formula and draw the graph. Use geometry.

7. ## Re: Find exact perpendicular distance between two parallel lines

Hi roger1505,

1 graph the lines
2 erect a vertical line @x=1. Apoint (1,1) on line y=3x-2 is produced
3@(1,1) erect a perpendicular to y=3x+3
4 write an equation for the perpendicular
5 solve for intersection of 4 and y=3x +3
6 use distance formula to find question d
d^2 = delta y^2 + delta x^2 (slope diagram between (1,1) and solution of 5 above

There is a simple solution using trig and the slope angle of the parallel lines

8. ## Re: Find exact perpendicular distance between two parallel lines

Originally Posted by roger1505
i wasnt in class when they taught this, but does anyone know?
If $\displaystyle Ax+By+C=0$ is a line where $\displaystyle |A|+|B|\ne 0$ and $\displaystyle P(p,q)$ is a point then the distance from that point to the line is $\displaystyle \frac{|Ap+Bq+C|}{\sqrt{A^2+B^2}}~.$

9. ## Re: Find exact perpendicular distance between two parallel lines

Hello, roger1505!

If you are allowed to use Trigonometry, here is another solution.

Find the exact perpendicular distance between two parallel lines: .$\displaystyle \begin{array}{ccc}y &=& 3x-2 \\ y &=& 3x + 3 \end{array}$

Code:
        |
|   /
|  /        /
| /        /
|/        /
A *        /
: |@*     /
: |   *  /
: |     * C
5 |    /
: + - / - - - -
: |  /
: | /
: |/ @
B * - - - - E
|
The two lines have y- intercepts at 3 and -2, and identical slopes.
$\displaystyle AB \,=\,5.$
Draw $\displaystyle AC$ perpendicular to the line through $\displaystyle B.$

Let $\displaystyle \theta = \angle CBE$
Then the slope of $\displaystyle BC \,=\,\tan\theta \,=\,3$
Note that $\displaystyle \angle CAB \,=\, \theta.$

We have: .$\displaystyle \tan\theta \:=\:\frac{3}{1}\:=\:\frac{\text{opp}}{\text{adj}}$
$\displaystyle \theta$ is in a right triangle with: $\displaystyle opp = 3,\;adj = 1$
Hence: .$\displaystyle hyp \,=\, \sqrt{10} \quad\Rightarrow\quad \cos\theta \,=\,\tfrac{1}{\sqrt{10}}$

In right triangle $\displaystyle ACB\!:\;\;\cos\theta \,=\,\frac{AC}{5} \quad\Rightarrow\quad AC \:=\:5\cos\theta \:=\:5\left(\tfrac{1}{\sqrt{10}}\right)$

Therefore: .$\displaystyle AC \:=\:\frac{\sqrt{10}}{2}$

10. ## Re: Find exact perpendicular distance between two parallel lines

Originally Posted by roger1505
Find the exact perpendiuclar distance between two parallel lines.
a)y=3x-2 and y=3x+3
As long as we are into spoon feeding, look at reply #8.
Rewrite line #1 as $\displaystyle 3x-y-2=0$. Note that $\displaystyle (1,6)$ is on $\displaystyle y=3x+3$.

Apply the formula from reply #8: $\displaystyle \frac{|3(1)-1(6)-2|}{\sqrt{(3)^2+(-1)^2}}=\frac{5}{\sqrt{10}}=\frac{\sqrt{10}}{2}.$

11. ## Re: Find exact perpendicular distance between two parallel lines

$\frac{|3(1)-1(6)-2|}{\sqrt{(3)^2+(-1)^2}}=\frac{5}{\sqrt{10}}=\frac{\sqrt{10}}{2}.$

This is the best and simplest way of finding the perpendicular distance between two parallel lines. Hope you get it right.