Coordinate plane: A two-dimensional surface on which points are plotted and located by their x and y coordinates.
The coordinate plane is a two-dimensional surface on which we can plot points, lines and curves. It has two scales, called the x-axis and y-axis, at right angles to each other. The plural of axis is 'axes' (pronounced "AXE-ease"). Points on the plane are located using two numbers - the x and y coordinates. These are the horizontal and vertical distances of the point from a specific location called the origin.
The horizontal scale is called the x-axis. As you go to the right on the scale from zero, the values are positive and get larger. As you go to the left from zero, they get more and more negative.
The vertical scale is called the y-axis. As you go up from zero the numbers are increasing in a positive direction. As you go down from zero they get more and more negative.
Along each axis you will see small tic marks with numbers. These are labels to help judge the scale. They are shown every unit in the figure above, but can be any increment, and need not be the same on both axes.
The point where the two axes cross (at zero on both scales) is called the origin. The origin is the point from which all distances along the x and y axes are measured.
The two axes divide the plane into four areas called quadrants. The first quadrant, by convention, is the top right, and then they go around counter-clockwise. In the diagram above they are labelled Quadrant 1,2 etc. It is conventional to label them with numerals but we talk about them as "first, second, third, and fourth quadrant". They are also sometimes labelled with Roman numerals: I, II, III and IV.
Things you can do in Coordinate Geometry
If you know the coordinates of a group of points you can:
- Determine the distance between them
- Find the midpoint, slope and equation of a line segment
- Determine if lines are parallel or perpendicular
- Find the area and perimeter of a polygon defined by the points
Determine the distance between points
Given the coordinates of two points, the distance between the points is given by:
where x2-x1 is the difference between the x-coordinates of the points.
and y2-y1 is the difference between the y-coordinates of the points.The formula above can be used to find the distance between two points when you know the coordinates of the points . This distance is also the length of the line segment linking the two points. This formula is simply a use of Pythagoras' Theorem.
The line segment AB is the hypotenuse of a right triangle, where one side is the difference in x-coordinates, and the other is the difference in y-coordinates (sometimes the difference in x coordinates is expressed as “dx” and the difference in y coordinates as “dy”). From Pythagoras' Theorem we know that AB2 = (x2-x1)2 + (y2-y1)2
Solving this for AB gives us the formula:
Lines in Coordinate Geometry
How to define a line
Consider the line in Fig 1. How would I define that particular line? What information could I give you over the phone so that you could draw the exact same line at your end?
Fig 1. How to define this line?
There are three ways commonly used in coordinate geometry:
- Give the coordinates of any two points on the line
- Give the coordinates of one point on the line, and the slope of the line
- Give an equation that defines the line.
It does not matter whether we are talking about a line, ray or line segment. In all cases any of the above three methods will provide enough information to define the line exactly.
(1) Using two points
In Fig 2, a line is defined by the two points A and B. By providing the coordinates of the two points, we can draw the line. No other line could pass through both these points and so the line they define is unique. I could call you on the phone and say "Draw a line through (9,9) and (17,4)" and you could reconstruct it perfectly on your end.
Fig 2. A,B define a unique line
(2) Using one point and the slope
Fig 3. Point and slope define the line
The other common method is the give you the coordinates of one point and the slope of the line. For now, you can think of the slope as the direction of the line. So once you know that a line goes through a certain point, and which direction it is pointing, you have defined one unique line.
In Fig 3, we see a line passing through the point A at (14,23). We also see that its slope is +2 (which means it goes up 2 for every one across). with these two facts we can establish a unique line.
The value of the slope is usually denoted by the letter m. For more on slope and how to determine it see Slope of a Line.
(3) Equation of a line
Once you have defined a line using the point-slope method, you can write algebra equations that describe the line. By applying algebraic processes to these equations we can solve problems that are otherwise difficult. These and many other graphing techniques are covered in the algebra volume, but the general idea is described here in Coordinate Geometry.
There are two types of equation commonly used to describe a line:
- Slope-intercept (the most common). Described in Equation of a line (Slope-Intercept)
- Point-slope. Described in Equation of a line (Point-Slope).
Both forms are really both variations on the same idea. In both cases you need to know the coordinates of one point, and the slope of the line.
- In the slope-intercept form, the given point is always on the y-axis and you supply the y-coordinate of that point (Its x-coordinate is always zero).
- In the point-slope form, you can use any point.
The place where the line crosses the y-axis is called the intercept, and is commonly denoted by the letter b. For more on this see Intercept of a line.
y = m(x-Px) + Py
Fig 4. Point-slope
y = mx + b
Fig 5. Slope-Intercept
If you look closely at the two formulae, you can see that they are quite similar. If you take the point-slope version in Fig 4 and choose to put P on the y-axis, its x coordinate is zero, and its y-coordinate is the same as the intercept. If you substitute those things you will get the slope-intercept formula on the right in Fig 5.
What are the equations used for?
- You can use them to actually plot the line: Take various values of x, and then use the equation to find the corresponding values of y. Plot the pairs to graph the line.
- If you know just one coordinate of a point on the line, you can find the other.
When two lines are parallel, they do not intersect anywhere. If you try to find the intersection, the equations will be an absurdity. For example the lines y=3x+4 and y=3x+8 are parallel because their slopes (3) are equal.
- Introduction to Coordinate Geometry http://www.mathopenref.com/coordintro.html
- Basic Geometry: Coordinate Plane https://www.khanacademy.org/math/basic-geo/basic-geo-coord-plane
- The Coordinate Plane http://www.math.com/school/subject2/lessons/S2U4L1GL.html
- The Coordinate Plane (very basic) https://www.mathplanet.com/education/algebra-1/visualizing-linear-functions/the-coordinate-plane
- Coordinate planes worksheets (scroll down halfway) http://www.commoncoresheets.com/Algebra.php
- Short, basic coordinate plane questions http://study.com/academy/practice/quiz-worksheet-the-coordinate-plane.html
- Ordered Pairs and Coordinate Plane Worksheets https://www.mathworksheets4kids.com/ordered-pairs.php
- Graphing Worksheets http://www.math-aids.com/Graphing/
- Points on the Coordinate Plane Example https://www.khanacademy.org/math/pre-algebra/pre-algebra-negative-numbers/pre-algebra-coordinate-plane/v/the-coordinate-plane
- Coordinate Plane and Plotting Points https://www.youtube.com/watch?v=r16I6LB2YbQ