# Find exact perpendicular distance between two parallel lines

• Jun 16th 2012, 05:18 PM
roger1505
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
• Jun 16th 2012, 05:54 PM
Plato
Re: Find exact perpendicular distance between two parallel lines
Quote:

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.
• Jun 16th 2012, 06:08 PM
roger1505
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
• Jun 16th 2012, 06:32 PM
Wilmer
Re: Find exact perpendicular distance between two parallel lines
Your teacher did not show how...or you missed classes?
• Jun 16th 2012, 06:53 PM
roger1505
Re: Find exact perpendicular distance between two parallel lines
i wasnt in class when they taught this, but does anyone know?
• Jun 16th 2012, 07:11 PM
richard1234
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.
• Jun 17th 2012, 07:30 AM
bjhopper
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
• Jun 17th 2012, 08:13 AM
Plato
Re: Find exact perpendicular distance between two parallel lines
Quote:

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 $\displaystyle \frac{|Ap+Bq+C|}{\sqrt{A^2+B^2}}~.$
• Jun 17th 2012, 03:12 PM
Soroban
Re: Find exact perpendicular distance between two parallel lines
Hello, roger1505!

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

Quote:

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

• Jun 17th 2012, 03:55 PM
Plato
Re: Find exact perpendicular distance between two parallel lines
Quote:

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: $\displaystyle \frac{|3(1)-1(6)-2|}{\sqrt{(3)^2+(-1)^2}}=\frac{5}{\sqrt{10}}=\frac{\sqrt{10}}{2}.$
• Aug 5th 2012, 11:09 PM
louisejane
Re: Find exact perpendicular distance between two parallel lines
http://latex.codecogs.com/png.latex?...%7D%7D%7B2%7D.

This is the best and simplest way of finding the perpendicular distance between two parallel lines. Hope you get it right. (Clapping)
• Aug 6th 2012, 05:28 AM
Wilmer
Re: Find exact perpendicular distance between two parallel lines