Vocabulary and representation in the coordinate plan
Vocabulary : points, plane, line, line segments and rays
Point
A point is a location. It does not have any size, and it is indicated by a dot.
Plane
A plane is a flat two dimensional surface extending infinitely far.
Planes are useful locations to study points on.
Line
A line is a straight one dimensional figure without thickness which extends infinitely far.
Line segment
A line segment is a portion of a line which connects two points.
Unlike a line that stretches out infinitely, a line segment has finite length.
Ray
A ray is a line with a single endpoint and extends infinitely in one direction.
Representation in the coordinate plane
The coordinate plane allows us to label points by their location.
The point in the following coordinate plane has an x -value of 2 and y -value of 3. We therefore label the point as \left(2{,}3\right).
Because lines, line segments, and rays are collections of points, they too can be represented in the cartesian plane.
The following graphic contains several points, lines, line segments, and rays in the cartesian plane.
Lines in the cartesian plane are defined by equations.
The following image is the graph of the linear equation y=2x+1. The graph is a line in the cartesian plane.
Midpoints, distance, lengths
Additive property of length
Suppose A,B,C are colinear points in the plane as illustrated in the following graphic. Then the length of the line segment adjoining A to C is equal to the sum of the lengths of the line segments adjoining A to B and B to C :
AC=AB+BC
Suppose that in the following graphic the length of the line segment from A to B is 7 and the length of the line segment from B to C is 5 :
Then the length of the line segment from A to C is:
AC=5+7=12
Midpoint formula
Midpoint
The midpoint of a line segment is the point which separates the line segment into two equal pieces.
In the following graphic, a line segment connecting the points A and B is given and the midpoint is labeled M.
Suppose the following are points in the Cartesian plane:
- A=\left(x_1,y_1\right)
- B=\left(x_2,y_2\right)
The midpoint of the line segment joining A and B is:
M=\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)
The midpoint of the line segment adjoining \left(1{,}2\right) and \left(5{,}6\right) is:
\left(\dfrac{1+5}{2},\dfrac{2+6}{2}\right)=\left(3{,}4\right)
The distance formula
Suppose the followings are points in the Cartesian plane:
- A=\left(x_1,y_1\right)
- B=\left(x_2,y_2\right)
The distance from points A to B is the length of the line segment adjoining A to B and is denoted d\left(A,B\right). It is equal to:
d\left(A,B\right)=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}
The distance between points A=\left(1{,}2\right) and B=\left(4{,}6\right) is:
d\left(A,B\right)=\sqrt{\left(4-1\right)^2+\left(6-2\right)^2}=\sqrt{25}=5
Parallel and perpendicular lines
Parallel and secant lines
Parallel lines
Two lines in the coordinate plane are parallel if they do not intersect.
Suppose the followings are two linear functions:
- y=m_1x +b_1
- y=m_2x+b_2
The graphs of these two linear functions are parallel if and only if:
- m_1=m_2 and
- b_1\not= b_2
The graphs of the following two linear equations are parallel:
- y=2x+1
- y=2x-7
Perpendicular lines
Perpendicular lines
Two lines in the coordinate plane are perpendicular if they intersect at a right angle.
Suppose the followings are two linear functions with m_1,m_2\not=0 :
- y=m_1x+b_1
- y=m_2x+b_2
The graphs of these two linear functions are perpendicular if and only if:
m_1=\dfrac{-1}{m_2}
The graphs of the linear functions y=2x+1 and y=\dfrac{-1}{2}x+3 are perpendicular.
