Theorems 1.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
Theorem 1.1: If (x1, y1) and (x2, y2) are any two points on the line whose equation is y = mx + b, then the slope between them is m.1
Theorem 1.2: A vertical line has infinite slope2. A line is not vertical if and only if its equation can be written in the form
Theorem 1.3: There is a uniquely determined line going through a given point with a given slope. If the given point is (x1, y1), and the line is vertical3, then the equation is
If the line is not vertical, then the equation can be written as
which can be simplified to
where
Theorem 1.4: Two points uniquely determine a line. If the points are (x1, y1) and (x2, y2) then the equation is
if the two points are vertical, and if not then it is of the form
where
and
Theorem 1.5: If (x1, y1) and (x 2, y2) are points on the line whose equation is y = mx + b then the distance between them isT1
Theorem 1.6: If two distinct lines are parallel, then they do not meet. If they are not parallel, then they meet at a uniquely determined point. If one of the lines is vertical, its equation will be of the form
If the other one is not vertical, it will have an equation of the form
and the point of intersection will be
If neither of the lines are vertical, they will have equations of the form
The coordinates of the point of intersection 4 are
and
Theorem 1.7: If we are given a line and a point not on the line, then there is a unique line, going through a given point, parallel to the given line.5
Theorem 1.8: If we are given a point and a line, there is a unique line, through the point, perpendicular to the given line.
Theorem 1.9: Given a line whose equation is
and a point whose coordinates are
the equation of the unique line through the point, perpendicular to the line, is
Theorem 1.10: Given a line whose equation is
and a point whose coordinates are
the coordinates of the foot of the point in the line are
Theorem 1.11: The perpendicular distance from the point
to the line
is
Theorem 1.12: Parallel lines stay the same distance apart. If the equations of the lines are
then the perpendicular distance from any point on one line to the other line isT2
If the lines are vertical, then their equations are
and the distance between the lines is
Theorem 1.13: A line is parallel to one of two parallel lines if and only if it is parallel to the other.
Theorem 1.14: If a line is perpendicular to one of two parallel lines then it is perpendicular to the other.
Theorem 1.15: If two lines are perpendicular to the same line then they are parallel.
Theorem 1.16: If a line is parallel to one of two perpendicular lines, it is perpendicular to the other.