Distance between two lines using slope intercept form

Question

One line; -x+y=0, and another line; y=x+3

I must find the distance between these points

 

Geometry

Answer 1

Graph the two lines

-x + y = 0

y = x

 

y = x + 3

 

The distance is along the perpendicular between the two lines. One such perpendicular line is y = -x. Graph it.

At what point does y = -x intersect y = x + 3?

Determine the distance from that point to (0, 0).

Answer 2

-x+y=0 and y=x+3

you want the perpendicular distance between the two lines

find the equation of one perpendicular line between the two parallel lines (they are parallel because they have the same slope)

y=x+3

y=x

draw the perpendicular line from the point (0,3) on line y=x+3

the slope of the perpendicular line is the negative reciprocal of one of the parallel lines, that is -1

the y-intercept is 3

the equation of the line is y=-x+3

using one of the parallel lines, y=x, and the perpendicular line, y=-x+3, you have a system of linear equations and you solve the system for (x,y), the point of intersection of the lines and therefore the other endpoint of the perpendicular line

y=x

y=-x+3

substitute

x=-x+3

x+x=3

2x=3

x=3/2

therefore y=3/2 also

now you have the endpoints of the perpendicular distance between the parallel lines

use the distance formula with...

(0,3) and (3/2,3/2)

√[(3-1.5)^2+(0-1.5)^2]

√[(1.5)^2+(1.5)^2]

√[(3/2)^2+(3/2)^2]

√[(9/4)+(9/4)]

√(18/4)

√18/√4

3√2/2 is the perpendicular distance between the two parallel lines

another solution...

because one line is y=x, draw a line from (0,3) to (3,0)

this forms a right isosceles triangle

using the Pythagorean Theorem...

3^2+3^2=P^2

9+9=P^2

18=P^2

P=√18

P=3√2, the hypotenuse of the triangle and the line that you drew

this line is half of the perpendicular distance between the parallel line

thus, 3√2/2 is the perpendicular distance between the two parallel lines

 

 

Answer 3

You have two equations of lines -x+y =0 , and y = x+3

 

by rearranging terms of the first equation you get y = x

 

the two equations above are of the form y = mx + b

 

where m is the slope of the line and b is the y intercept

 

The two lines have the same slope m =1, and therefore they are parallel

 

the y intercept of the first line y= x is 0 and the y intercept of the

2nd line y=x+3 is 3, with two parallel lines the vertical distance would be 3.

 

Another method you can use is find the coordinates of a point on each line and apply the slope intercept form. we know that the slope is 1 for both lines. the two points will be as follows:

 

y=x, for x =1 , y=1 and the point is (1,1)

y=x+3 for x =1 y = 3+1 = 4 and the point is (1,4)

 

The distance between the two points D = √(y2-y1)²+(x2-x1)²

D=√(4-1)²+(1-1)², D = √9=3 same as above as d is the vertical distance between the two points with the same x coordinate