find the distance between the lines with the equations y=(2/7)x+4 and y= (2/7)-2

Question

Please explain how you solved this.

Answer 1

We need to find the line that is perpendicular to both these parallel lines.  Perpendicular lines have negative reciprocal slope.

 

m = -7/2

 

So your perpendicular line is now       y = (-7/2)x + 4

 

This line is perpdicular to the first line.  The point of intersection is (0, 4).  You will use this point later.

 

Now we set this line equal to the last line to find the point of intersection.

 

 

-(7 / 2)x + 4 = (2 / 7)x - 2

 

Solve for x from this equation.

 

 

Multiply both sides of the equation by 2.

 

-7x + 8 = (4 / 7)x - 4

 

 

Multiply both sides of the equation by 7.

 

-49x + 56 = 4x - 28

 

-49x - 4x = -28 - 56

 

-53x = -84

 

     x₂ = 1.585

 

 

Plug in this value of x into any side to solve for y.

 

y₂ = (2 / 7)(1.585) - 2

 

 

This is your second point of intersection: (x₂ , y₂).

 

Finally, find the distance between the two points of intersection using the distance formula.  That will be your distance between the two given lines.

Answer 2

As written, your two lines intercept, so I'm going to assume that the second equation should be y = (2/7)x-2.  So, we're looking for the distance between these two along a perpendicular line.  If we arbitrarily pick a point on the first line (0,4) and draw a perpendicular line from it to the second line, let's call the point it intercepts A.  Let's call that distance x.  If we draw a vertical line from (0,4) down to the second line (0,-2) it has length 6.  Based on similar triangles, we can observe that the distance between (0,-2) and A will be (2/7)x.  Therefore, we can apply Pythagorean theorem to find:

 

x² + (2/7 * x)² = 6²

x² * (1 + (2/7)²) = 36

x² * (53/49) = 36

x² = 36*49/53

x = 6*7/√53 = 5.77

 

Note that you could alternatively derive the equation for a perpendicular line through (0,4) and solve for where that intercepts the second line, then assess the distance between those two points.  There are, of course, lots of ways to skin a cat.