SOLUTION: Find the distance between two parallel lines Y=2x-1, y=2x+9 I tried graphing it and finding the perpendicular intercept but my numbers are not coming out right.

Algebra ->  Length-and-distance -> SOLUTION: Find the distance between two parallel lines Y=2x-1, y=2x+9 I tried graphing it and finding the perpendicular intercept but my numbers are not coming out right.      Log On


   

Question 1179087: Find the distance between two parallel lines
Y=2x-1, y=2x+9
I tried graphing it and finding the perpendicular intercept but my numbers are not coming out right.
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The video referenced by the other tutor shows a perfectly good way of solving the problem:
(1) Using the slopes of the given lines and y-intercept of the first line, find the equation of the line perpendicular to that first line containing that y-intercept;
(2) use the equation of that perpendicular line and the equation of the second given line to find the point of intersection; and
(3) use the distance formula with that point of intersection and the y-intercept of the first line to get the distance between the two lines.

There is a lot of good math in that solution process; you should understand it and know how to use it.

But there are much easier and faster ways to answer the problem. Below are two of them.

First alternative: Make a sketch with a right triangle

Sketch a graph of the two lines with slope 2 and y-intercepts -1 and +9;
Draw the perpendicular segment from (0,9) to the other line;
Look at the right triangle whose sides are that perpendicular segment, part of the second line, and part of the y-axis. Since the slopes of the two lines are 2, we can call the lengths of the two legs of that right triangle x and 2x; and the hypotenuse is 10. Then from the Pythagorean Theorem,

x%5E2%2B%282x%29%5E2=10%5E2
x%5E2%2B4x%5E2=100
5x%5E2=100
x%5E2=20
x+=+2%2Asqrt%285%29

ANSWER: The distance between the two lines (the length of the perpendicular segment between the two lines) is x = 2*sqrt(5)

Second alternative: Use the point-to-line distance formula

A very useful formula to know is for finding the distance from a given point to a given line:

Given the equation of a line in the form Ax+By+C=0 and a point (p,q), the distance from the point to the line is

abs%28%28Ap%2BBq%2BC%29%2F%28sqrt%28A%5E2%2BB%5E2%29%29%29

Use the equation of the first line in the required form (2x-y-1=0) and the y-intercept of the second line (0,9) as the fixed point and plug the numbers into the formula: