Two supports beams are modeled by lines y=2x+10 and y=2x+15. What is the distance between two lines?

Question

I have no idea how to do this 

Answer 1

The two lines given are parallel to each other because they have the same slope.  So to find the distance between the parallel lines, we need to find the line that is perpendicular to both lines.  Perpendicular lines have negative reciprocal slopes.  So the slope of the perpendicular line is -1/2.

 

Suppose the perpendicular lines intersects at either of these 2 lines.  Lets use the line y=2x+10.  To find the equation of the perpendicular line, we use the point-slope form of a line

 

y = m(x - x₁) + y₁

 

where:

m = -1/2

(x₁ , y₁) = (0, 10)  ---> any point on the line  y=2x+10

 

 

So now our equation is

 

y = (-1/2)x + 10

 

 

Next, we find the point of intersection between this perpendicular line and the other line y=2x+15.

 

2x + 15 = (-1/2)x + 10

 

 

Solve for x from this equation.

 

Multiply both sides of the equation by -2.

 

-4x - 30 = x - 20

 

-4x - x = -20 + 30

 

-5x = 10

 

  x = -2

 

 

Plug in this value into the line   y=2x+15   to solve for y.

 

y = 2(-2) + 15

 

y = -4 + 15

 

y = 11

 

 

So now your second point is (-2, 11).  Finally, you will find the distance between (0, 10) and (-2, 11) to find the distance between the two parallel lines. 

 

In the graphical sense, if you were to connect these two points, you will get a line the is perpendicular to the two parallel lines given.

 

 

Answer 2