Section 6 : Perpendiculars and Distance
Distance Between a Point and a Line:
The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.
The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.
Perpendicular Postulate:
If given a line and a point not on the line, then there exists exactly one line through the point that is perpendicular to the given line.
Find the distance from R to S.
Line R contains points ( 2,2) and (4,8) . Point S has coordinates (4,2)
First find the slope and y- intercept of line R and write the equation for the line.
m= y2- y1 8- 2 6
-------- = ------- = ---- or 3
x2- x1 4-2 2
Since R contains ( 2,2), the y- intercept is 3.
So, the equation for the line R is y= 3x+2. The slope of the line perpendicular to R, line S, is -3. Write the equation of line S through (4,2) with slope -3.
y= mx+b slope- intercept formula
2= -3(4) +b m=-3 (x,y) = (4,2)
2= -12 +b simplify
14= b add 12 each side
so, the equation for line S is y= -x+(14). Solve the system of the equations to determine the point of the intersection.
line R: y= x+2
line S: (+) y= -x+14 add the two equation and divide by 2.
-----------------
2y= 16
y= 8
solve: for x.
y= x+2
8= x+2 subtract 2 each side.
6 =x
The point of intersection is (-4,-2). Let this be point T. Use the Distance Formula to determine the distance between S (4,2) and T (8,6).
Line R contains points ( 2,2) and (4,8) . Point S has coordinates (4,2)
First find the slope and y- intercept of line R and write the equation for the line.
m= y2- y1 8- 2 6
-------- = ------- = ---- or 3
x2- x1 4-2 2
Since R contains ( 2,2), the y- intercept is 3.
So, the equation for the line R is y= 3x+2. The slope of the line perpendicular to R, line S, is -3. Write the equation of line S through (4,2) with slope -3.
y= mx+b slope- intercept formula
2= -3(4) +b m=-3 (x,y) = (4,2)
2= -12 +b simplify
14= b add 12 each side
so, the equation for line S is y= -x+(14). Solve the system of the equations to determine the point of the intersection.
line R: y= x+2
line S: (+) y= -x+14 add the two equation and divide by 2.
-----------------
2y= 16
y= 8
solve: for x.
y= x+2
8= x+2 subtract 2 each side.
6 =x
The point of intersection is (-4,-2). Let this be point T. Use the Distance Formula to determine the distance between S (4,2) and T (8,6).
Distance Formula:
x1=4, y1=2 x2= 8, y2= 6
simplify
so, the distance between the lines is square root of 32.
simplify
so, the distance between the lines is square root of 32.
Equidistant: the distance between two lines measured along a perpendicular line is always the same.
Parallel line do not intersect. The distance between two parallel lines is the perpendicular distance between one of the line and any point on the other line.
Theorem 3.9: In a plane, if two lines are each equal distance from the third line then the two line are parallel from each other.
Parallel line do not intersect. The distance between two parallel lines is the perpendicular distance between one of the line and any point on the other line.
Theorem 3.9: In a plane, if two lines are each equal distance from the third line then the two line are parallel from each other.