the line y =2x +7 is parallel to the line y=2x-3.The line y=-0.5x+7 is perpendicular to the line y= 2x-3.Calculate the perpendicular distance between the pair of parallel lines.

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nick-teal | High School Teacher | (Level 3) Adjunct Educator

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The reason they give you the perpendicular line is so that you can find the points where that perpendicular line intersects each of the parallel lines, and then use the distance formula between those two points to find the distance between the lines.

`-.5x + 7 = 2x + 7`

`x = 0`

The first point is (0, 7)

`-.5x + 7 = 2x - 3`

` `

`x = 4`

The second point is (4, 5)

Using distance formula

`sqrt((4 - 0)^2 + (5 - 7)^2)`

`sqrt(20)`

`2sqrt(5)`

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

Posted on

Actually, to calculate distance between two parallel lines their equations are sufficient.

If we have two lines in the form
ax + by = c1 and ax+by = c2,
then the distance is |c1-c2|/sqrt(a^2+b^2).

Here a=2, b=-1, c1=-7, c2=3. So the answer is |-7-3|/sqrt(2^2+1^2) = 10/sqrt(5) = 2*sqrt(5). It is approx.= 4,472.

if we have to use this perpendicular line, then find the intersections:
1) y=2x+7, y=-0.5x+7.
2x+7=-0.5x+7, 2x=-0.5x, x=0. Then y=7.

2) y=2x-3, y=-0.5x+7.
2x-3=-0.5x+7, 2.5x=10, x=4. Then y=5.

The distance between (0, 7) and (4, 5) is
sqrt(4^2 + 2^2) = sqrt(20) = 2*sqrt(5).

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