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# Geometry Problem: Find distance between parallel lines

Find distance between y = -2x + 3 and y = -2x - 3 (Hint: use (0,3)

### 2 Answers by Expert Tutors

David W. | Experienced ProfExperienced Prof
4.4 4.4 (47 lesson ratings) (47)
1
The distance between two parallel lines ranges from the shortest distance (two intersection points on a perpendicular line) to the horizontal distance or vertical distance to an infinite distance.

The vertical distance between the two given parallel lines is from the point (0,3) to the point (0,-3) [the two y-intercepts], which is 6. You may also compute the horizontal distance from (0,3) to (-3,3), which is 3. Now, these three points form a right triangle whose hypotenuse is divided by the perpendicular line, which intersects this hypotenuse, forming right triangles with a hypotenuse of 3 (horizontal) and a hypotenuse of 6 (vertical). This is why drawing a picture is very important.

To continue:

The line that is perpendicular to these parallel lines has a slope of (1/2) [that is, the negative reciprocal of (-2)], so the equation of that line has the form: y = (1/2)x + b and we need a point to determine b.
3 = (1/2)(0) + b
3 = b
The equation of the perpendicular line is: y = (1/2)x + 3

Check: The given y-intercept, (0,3), is on the perpendicular line.

The perpendicular line y=(1/2)x+3 intersects the other parallel line (y=-2x-3) when:
(1/2)x+3 = -2x-3
x+6 = -4x-6           [multiply both sides by 2]
5x = -12
x = -12/5
and y=(1/2)x+3
y=(1/2)(-12/5)+3     [insert value for x]
y=(-6/5)+(15/5)      [common denominator]
y=(9/5)

So, the two intersection points are (0,3) and (-12/5, 9/5).

The distance formula is:
d = √( (x2-x1)2 + (y2-y1)2 )
d = √( (-12/5-0)2 + (9/5-3)2 )
d = √( 144/25 + (9/5-15/5)2)
d = √( 144/25 + (-6/5)2)
d = √(144/25 + 36/25)
d = √(180/25)
d = √(36*5/25)
d = (6/5)√(5)
d = 2.68    (rounded)
Robert D. | Math/MATLAB/Electrical Engineering TutorMath/MATLAB/Electrical Engineering Tutor
4.9 4.9 (65 lesson ratings) (65)
0
The two lines are parallel i.e. slope is same!!!!

so the distance is 3-(-3)=6