﻿ Line Geometry
Line Geometry
 (1) Line equation: y = x +
 Two points on line: x1: y1: x2: y2:
 Angle (α): Intercepts:   (xi,0): (0,yi): Midpoint: xd: yd: Line length:
 Point in plane: xp:    yp: The point is on the line
Distance from point (xp, yp) to line (1):
Equation of line passing through (xp, yp)
and is perpendicular to line (1):
Intersection coordinates:
 x: y:
 Line basic definition ▲  The most general equation of a line is of the form: (1)
where  A, B  and  C  are any real number and  A  and  B  are not both zero. If  B ≠ 0  then we can divide
 equation (1)  by  B  to obtain the form: (2)
 This is the equation of a line whose slope is: and Equation (2) can be normalized to the general form: (3)
 the slope of the line (m) is defined in terms of the inclination (4)
Note: if the angle α is greater then 90 degree then the slope is negative.
 α (0 - 90) degree   :positive slope α (90 - 180) degree   :negative slope
Necessary condition for two lines to be perpendicular to each other is that their slopes fulfills the
 condition: m1 m2 = − 1 (5)
In order to find the intersection point of two lines we have to solve the system of linear equations
 representing the lines. A x + B y = −C D x + E y = −F
 solution by matrix:  If the determinant: then intersection point exists.
 Basic line of the form     y = ax + b     or     Ax + By + C = 0 ▲
Slope (m) of the line
 m = a   yintercept (yi)
 yi = b yi = −mxi yi = y1 − mx1 xintercept (xi)    tan θ
 tan θ = m  Line angle (θ)
from x axis (range 0 ≤ θ < π)
 θ = arctan(m)  Slope (M) of a line perpendicular to a given slope (m)    Line midpoint   Point (x, y) which divides the line connecting two points (x1 , y1) and (x2 , y2) in the ratio p:q  Point (x, y) which divides the line connecting two points (x1 , y1) and (x2 , y2) externally at a ratio p:q  Note: the (x,y) point is in the direction from point 1 to point 2,to get the other side extension change the point 1 with point 2 and vice versa
A point (x, y) which is located at a distance d from a point (x1 , y1) on the line  Angle θ between two lines:   Angle between two lines given by
 Ax + By + C = 0 Dx + Ey + F = 0  Lines equations ▲
Equation of a line passing through a point (x1 , y1)
 y = mx + (y1 − mx1) y − y1 = m(x − x1)
Equation of a line passing through two points (x1 , y1), (x2 , y2)  Equation of a line perpendicular to a given slope m and passing through a point (xp , yp)  Equation of a line perpendicular to a line which is defined by two points (x1 , y1) and (x2 , y2) and passing through the point (xp , yp)  Equation of a line passing through the intercepts xi , yi xiy = −yix + xiyi
Equation of a line passing through the point (xp ,yp) and parallel to a line which is defined by two points (x1 , y1) and (x2 , y2)  Equation of a line parallel to the line Ax + By + C = 0 and at a distance d from it. Equation of the midline between the lines    Ax + By + C = 0
Dx + Ey + F = 0 Equation of a line perpendicular to the line    Ax + By + C = 0
 Bx − Ay + C = 0
Equation of a horizontal line y = b
Equation of a vertical line x = a
 Lines distances     y = ax + b    Ax + By + C = 0 ▲
Distance between two points (D) Distance between intercepts xi and yi Distance from a line to the origin  Distance from a line given by two points (x1 ,y1),(x2 , y2) to the origin Distance from a line to the point
(xp , yp)  Distance from a line given by two points (x1 , y1) , (x2 , y2) to the point (xp , yp) Distance between two parallel lines
 y = ax + b y = cx + d
or
 Ax + By + C = 0 Dx + Ey + F = 0  