The lines 3x+ 4y= 12 and 3x+ 4y= 72 are parallel. Find the distance that separates these lines.

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- Mar 14th 2011, 06:51 PMthamathkid1729Distance Between 2 Parallel LinesThe lines 3
*x*+ 4*y*= 12 and 3*x*+ 4*y*= 72 are parallel. Find the distance that separates these lines.

- Mar 14th 2011, 07:12 PMProve It
What is the difference between the intercepts?

- Mar 15th 2011, 03:42 AMearboth
I assume that you mean the perpendicular distance between the two parallel lines.

1. Line has the y-intercept .

Line has the y-intercept .

2. The (perpendicular) distance of a point to a straight line with the equation is calculated by

3. Calculate the distance of P(0, 3) to :

4. So the distance you are looking for has the value 12. The negative sign of d indicates that the point P and the origin are on the same side of the straight line. - Mar 15th 2011, 04:31 AMHallsofIvy
Oh, blast! I had just claimed that those lines are NOT parallel, then realized that the "3" multiplying x in the first equation, at least on my web viewer, had been separated by a line break!

Here's how I would do that- because I am terrible at memorizing formulas! The slope of the two lines is -3/4 so a line perpendicular to both will have slope 4/3. The equation of a line with slope 4/3 and passing through (0, 3) is . That will cross the second line where . Then and . And, of course, .

That is, the line perpendicular to both given lines intersects them at and . I have left the y value in that form because we need to subtract 3 from it to find the distance between those points.

The distance is .

(Added after thinking it over) However, note Prove It's post. The first line passes through the point (0, 3) and the second through (0, 18). The difference between the two y intercepts is 18- 3= 15. Now that is NOT the distance between the lines (which is always measured perpendicular to both lines) but if we were to drop a perpendicular from, say, (0, 3) to the second line, it is the hypotenuse of a right triangle that has the common perpendicular to both lines as a leg. Further, a little geometry shows that those two legs in the ratio of 3 to 4 since the slope of both lines is -3/4. That is, if the perpendicular distance is "x" then the other leg has length (3/4)x so we must have or ..

. - Mar 15th 2011, 12:09 PMbjhopperparallel lines
Without getting involved in a lenghty response this problem can be solved quickly using similar triangles of 3,4,5 right triangles where the one defining the lenght of the perpendicular has a hypot of 15

bjh - Mar 15th 2011, 12:25 PMPlato