Things you can do with 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.
 Transform a shape by moving, rotating and reflecting it.
 Define the equations of curves, circles and ellipses.
Distance Between Two Points (given their coordinates)
Given the coordinates of two points, the distance D between the points is given by:
where dx is the difference between the x-coordinates of the points and dy is the difference between the ycoordinates 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.
The distance formula is simply a use of Pythagoras' Theorem.
Notice, AB is the hypotenuse of a right triangle, where one side is dx (the difference in x-coordinates) and the
other is dy (the difference in y-coordinates).
From Pythagoras' Theorem we know that: AB2 = dx2 + dy2
Solving this for AB gives us the formula:
Vertical and Horizontal Lines
If the line segment is vertical or horizontal, the formula above will still work, but there is an easier way. For a
horizontal line, its length is the difference between the x-coordinates. For a vertical line its length is the
difference between the y-coordinates.
Things you can do with Coordinate Geometry
A Practical Application
Interactive programs make extensive use of coordinate geometry. The computer screens you look at are grids
of thousands of tiny dots called pixels that together make up images. Each pixel is addressed using its (x, y)
coordinates. Each pixel has a unique pair of coordinates. Programmers use coordinate geometry to code
images on the computer screen.
Midpoint of a Line Segment
Also known as the Midpoint Theorem:
The coordinates of the midpoint of a line segment are the average of the coordinates of its endpoints.
 x  x y  y2 
The midpoint can be found by using the following formula: midpoint   1 2 , 1

2 
 2
A line segment on the coordinate plane is defined by two endpoints whose coordinates are known. The
midpoint of this line is exactly halfway between these endpoints and its location can be found using the
Midpoint Theorem, which states:
 The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints.
 Likewise, the y-coordinate is the average of the y-coordinates of the endpoints.
Along the x-axis: midway between 10 and 50 is
30.
Along the y-axis: midway between 10 and 20 is
15.
Look at this graphically in the figure above. Notice that on each axis, the black pointers from the midpoint C
are always exactly halfway between the orange pointers from the endpoints A and B.
Slope of a Line
The slope of a line is a number that measures its "steepness", usually denoted by the letter m.
It is the change in y for a unit change in x along the line.
The slope of a line (also called the gradient of the line, the rate of change…) is a number that describes how
"steep" it is.
Things you can do with Coordinate Geometry
In the figure above, notice that for every increase of one unit to the right along the horizontal x-axis, the line
moves down a half unit.
1
It therefore has a slope of - .
2
To get from point A to B along the line, we have to move to the right 30 units and down 15.
Again, this is a half unit down for every unit across.
Because the line slopes downwards to the right, it has a negative slope. As x increases, y decreases.
If the line sloped upwards to the right, the slope would be a positive number and we would say that as x
increases y also increases.
Formula for the Slope
Given any two points on the line, its slope is given by the formula where:
Ax
Ay
Bx
By
the x-coordinate of point A
the y-coordinate of point A
the x-coordinate of point B
the y-coordinate of point B
y2
y1
x2
x1
the y-coordinate of point B
the y-coordinate of point A
the x-coordinate of point B
the x-coordinate of point A
OR
y y
m 2 1
x2  x1
It does not matter which point you choose for A or B. As long as they are both on the line somewhere and they
are used consistently through the whole calculation, the formula will produce the correct slope.
Example
Refer to the previous diagram, substituting the coordinates for A and B into the formula, we get:
20  5
15
1
slope 


10  40
30
2
Things you can do with Coordinate Geometry
Finding the Slope by Inspection
Rather than just plugging numbers into the formula above, we can find the slope by understanding the
concept and reasoning it out.
Refer to the line on the right, defined by two given points A, B. We can see
that the line slopes up and to the right so the slope will be positive.
1. Calculate dx, the horizontal distance from the left point to the right
point. Since B is at (15, 5) its x-coordinate is the first number, 15. The xcoordinate of A is 30. So the difference (dx) is 15.
2. Calculate dy, the amount the line rises or falls as you go to the right.
Since B is at (15, 5) its y-coordinate is the second number or 5. The ycoordinate of A is 25. So the difference (dy) is 20.
It is positive because the line goes up as you go to the right. It would have been negative otherwise.
20 4

3. Dividing the rise (dy) by the run (dx): slope 
15 3
A way to remember this method is "rise over run".
It is the "rise" - the up and down difference between the points, over the "run" - the horizontal run between
them.
Just remember that rise going downwards is negative.
Slope Direction
The slope of a line can be positive, negative, zero or undefined.
Positive Slope
Here, y increases as x increases, so the line slopes upwards to the right. The slope will be a positive number.
3
3
The line above has a slope of about
, it goes up about
for every step of 1 along the x-axis.
10
10
Negative Slope
Here, y decreases as x increases, so the line slopes downwards to the right. The slope will be a negative
number.
3
3
The line above has a slope of about  , it goes down about
for every step of 1 along the x-axis.
10
10
Zero Slope
Here, y does not change as x increases, so the line in exactly horizontal.
The slope of any horizontal line is always zero.
Things you can do with Coordinate Geometry
The line above goes neither up nor down as x increases, so its slope is zero.
A horizontal line has an equation of the form y = b, where b is the y-intercept.
Undefined Slope
When the line is exactly vertical, it does not have a defined slope.
The two x-coordinates are the same, so the difference is zero.
#
The slope calculation is then something like; slope   UNDEFINED (uh-oh, can’t do that!)
0
When you divide anything by zero the result has no meaning.
The line above is exactly vertical, so it has no defined slope. We say "the slope of the line AB is undefined".
A vertical line has an equation of the form x = a, where a is the x-intercept.
Equation of a Line
The slope (m) of a line is one of the elements in the equation of a line when written in the "slope-intercept"
form: y  mx  b . The m in the equation is the slope of the line.
Slope as an Angle
The slope of the line can also be expressed as an angle, usually in degrees or radians.
Look at the figure above, by convention the angle is measured from any horizontal line (parallel to the x-axis).
Lines with a positive slope (up and to the right) have a positive angle, and a negative angle for a negative
slope.
To convert from slope m to slope angle and back:
angle = arctan(m)
m = tan(angle)
Parallel Lines
Two lines are parallel if they have the same slope, or if they are both vertical.
Things you can do with Coordinate Geometry
When two straight lines are plotted on the coordinate plane, we can tell if they are parallel from their slope. If
the slopes are the same then the lines are parallel.
The slope can be found using any method that is convenient to you:

y  y1 
 From two given points on the line.  m  2

x2  x1 

 From the equation of the line in slope-intercept form. [ y  mx  b ]

From the equation of the line in point-slope form. [ y  y1  m  x  x1  OR y  m  x  x1   y1 ]
When they are Vertical
Recall that if a line is vertical it has no defined slope. Vertical lines are parallel by definition.
A line is vertical if the x-coordinates of two points on the line are the same.
Example: Are two lines parallel?
In the figure above there are two lines. One line is defined by two points at (5, 5) and (25, 15). The other is
13
x  2.5 . Decide if they are parallel.
defined by an equation in slope-intercept form y 
25
For the top line, the slope is found using the coordinates of the two points that define the line.
15  5 10 1
slope 


25  5 20 2
For the lower line, the slope is taken directly from the formula. Recall that the slope-intercept formula is
13
y  mx  b , where m is the slope. So looking at the formula we see that the slope is
.
25
1
13
The top line has a slope of , the lower line slope is
, which are not equal. Therefore, the lines are not
2
25
parallel. The lines are very close to being parallel, and may look parallel, but appearance can deceive.
Things you can do with Coordinate Geometry
Example: Define a line parallel to a given line.
In the figure below AB is defined by two points. Plot a line through the given point C parallel to AB .
First find the slope of the AB using the same method as the example above.
20  7
13
13
slope 


10  35 25
25
For the line to be parallel to AB it will have the same slope, and will pass through a given point, C(12, 10). We
therefore have enough information to define the line by its equation in point-slope form:
13
y    x  12   10
25
If we wanted to go ahead and actually plot the line we can do so by finding another point on the line using the
equation and then draw the line through the two points.
Rectangle
A quadrilateral where all interior angles are 90°, and whose location on the coordinate plane is determined by
the coordinates of the four vertices (corners).
A rectangle is placed in the coordinate plane with each of the four vertices (corners) having
known coordinates. From these coordinates, various properties such as width, height etc. can be found.
 Opposite sides are parallel and congruent.
 The diagonals bisect each other.
 The diagonals are congruent.
Dimensions of a Rectangle
The dimensions of the rectangle are found by calculating the distance between various corner points.
Things you can do with Coordinate Geometry
In the figure above:
 The height of the rectangle is the distance between A and B (or C to D).
 The width is the distance between B and C (or A to D).
 The length of a diagonal is the distance between opposite corners, say B and D (or A to C since the
diagonals are congruent).
This method will work even if the rectangle is rotated on the plane, as in the figure above.
But if the sides of the rectangle are parallel to the x and y-axes, then the calculations can be a little easier.
 The height is the difference in y-coordinates of any top and bottom point - for example A and B.
 The width is the difference in x-coordinates of any left and right point - for example B and D.
Example
 The height of the rectangle is the distance between the points A and B. (Using C and D will produce the
same result). Use the formula for the distance between two points;
height 

2
2
 16.1
The width is the distance between the points B and C. (Using A and D will produce the same result).
Using the formula for the distance between two points;
width 

16  8  34  20
 47 16  34 16
2
2
 35.8
The length of a diagonals is the distance between B and D. (Using A and C will produce the same
result). Using the formula for the distance between two points;
diagonal 
39 16  34  2
2
2
 39.4
Rectangle Area and Perimeter
The area and perimeter of a rectangle can be found given the coordinates of its vertices (corners).
Things you can do with Coordinate Geometry
Area
In coordinate geometry, the area of a rectangle is calculated in the usual way once the width and height are
found. Once the width and height are known the area is found by multiplying the width by the height in the
usual way.
The formula for the area is: Area = width  height
[A = lw or wh]
Perimeter
The perimeter of a rectangle (the total distance around the edge) is calculated in the usual way once the width
and height are found. Once the width and height are known the perimeter is found by adding twice the width
to twice the height to calculate the distance around the edge of the rectangle.
The formula for the perimeter is:
Perimeter = (2  width) + (2  height)
[P = 2w + 2h]
Example
Use the figure above.
 The height of the rectangle is the distance between the points A and B. (Using C and D will produce the
same result). This one is 16.
 The width is the distance between the points B and C. (Using A and D will produce the same result).
The one above is 35.
Area is the width times height, or 16  35 = 560 units2
Perimeter is twice the width plus twice the height or (2  16) + (2  35) = 102 units
Square
A 4-sided regular polygon with all sides equal, all interior angles 90° and whose location on the coordinate
plane is determined by the coordinates of the four vertices(corners).
From the coordinates of the four vertices, various properties such as width, height etc. can be found.
A square has the following properties:
 All four sides are congruent.
 Opposite sides are parallel.
 The diagonals bisect each other at right angles.
 The diagonals are congruent.
Things you can do with Coordinate Geometry
Dimensions of a Square
The dimensions of the square are found by calculating the distance between various corner points. Recall that
we can find the distance between any two points if we know their coordinates.


The length of each side of the square is the distance any two adjacent points (say AB, or AD)
The length of a diagonals is the distance between opposite corners, say B and D (or A and C since the
diagonals are congruent).
This method will work even if the square is rotated on the plane (see rectangle example). But if the sides of
the square are parallel to the x and y-axes, then the calculations can be a little easier.
Example
 The side length of the square is the distance between any two adjacent vertices. Let's pick B and C.
Since that side is horizontal, by inspection we can find the length to be 22.
 The length of a diagonal is the distance between any pair of opposite vertices. Using the distance
formula we can find the distance from B to D:
diagonal 
 29  7    4  26
2
2
 222   22  22 2
2
In a square, the diagonal is always the length of a side times the square root of two (from 45, 45, 90
special right triangles): 22 2
Area and Perimeter of a Square
The area and perimeter of a square can be found given the coordinates of its vertices (corners).
Things you can do with Coordinate Geometry
Area
The area of a square is calculated in the usual way once the length of a side is found. Once the side length is
known the area is found by multiplying the side length by itself in the usual way. The formula for the area is:
area = s2 where s is the length of any side (they are all the same).
The "Diagonals" Method to find Area
If you know the length of a diagonal, the area is given by:
where d is the length of either diagonal
The length of a diagonal can be found by using the distance formula to find the distance between opposite
vertices, say A and C in the figure.
Perimeter
A square has four sides which are all the same length. The perimeter of a square (the total distance around the
edge) is therefore the four times the length of any side. See square definition to see how the side length is
calculated.
The formula for the perimeter is P = 4s where s is the length of any side (they are all the same).
Example
Use the diagram given previously to calculate the area and perimeter, the example below assumes you know
how to calculate the side length of the square.
 The side length of the square is the distance between the points A and B. (Or any two adjacent
vertices). Here, this is 22.
 Area is the side length times itself, or 22 x 22 = 484
 Perimeter is four times the side length or 4 x 22 = 88
Trapezoid
A quadrilateral that has one pair of parallel sides, and where the vertices have known coordinates.
Things you can do with Coordinate Geometry
Things you can do with Coordinate Geometry
Altitude of a Trapezoid
In the previous figure, the altitude is the perpendicular distance between the two bases (parallel sides). To
find this distance, we can use the distance formula from a point to a line. For the point, we use any vertex, and
for the line we use the opposite base. In the figure above we have used the distance from point B to the
opposite base AD.
This method will work even if the trapezoid is rotated on the plane, but if the sides of the trapezoid are
parallel to the x and y-axes, then the calculations can be a little easier. The altitude is then the difference in ycoordinates of any point on each base, for example A and B.
Median of a Trapezoid
Recall that the median is a line segment linking the midpoints of the two legs of the trapezoid. (The legs are
the two non-parallel sides.) We can find the midpoint of a leg by using the midpoint formula. By applying this
twice, once for each leg, the median can be drawn between them.
The length of the median can be found in two ways:
1. The median length is the average of the two bases (parallel sides). Find the length of each base by
using the distance formula. Then find the average of these two lengths by adding them and dividing by
2.
2. Find the midpoints of the legs using the midpoint formula, and then find the distance between them
using the distance formula.
Examples
In the worked examples below, we will calculate the properties of the trapezoid in the previous figure.
 The altitude of the trapezoid.
Since in this case the bases (parallel sides) of the trapezoid are parallel to the x-axis, the altitude can be
found as the difference between the y-coordinates of any point on each base. Let's pick B and A. The ycoordinate of B is 31, and the y-coordinate of A is 7, so:
Altitude = 31–7 = 24
If the trapezoid had been rotated, then the altitude would be found using the distance formula, in this
case using the point A, and finding the distance to the line BC.
 The endpoints of the median are located at the midpoints of AB and CD, which can be found using the
midpoint formula. To find G, the midpoint of AB:
The x-coordinate of G is the average of the x-coordinates of A and B:
16  6
Gx 
 11
2
31  7
 19 and so, one end
and the y-coordinate is the average of the y-coordinates of A and B: G y 
2
of the median is at G(11, 19). We can use the same method to find the other end, point H is located at
H(42, 19).
 The length of the median is the distance between the midpoints G and H of AB and CD. Using the
distance formula, we see that in this case, the median is parallel to the x-axis, so the length is the
difference in the x-coordinates of G and H:
Median Length = 42–11 = 31
Things you can do with Coordinate Geometry
Trapezoid area and perimeter
The area and perimeter of a trapezoid can be calculated if you know the coordinates of its vertices.
Area
The area of a trapezoid is calculated by multiplying the average width by the altitude. (Note too that the
median length is the same as the average width.)
As a formula:
Area 
1
b b 
h  b1  b2  OR h  1 2 
2
 2 
Where b1, b2 are the lengths of the two bases (BC and AD) and
h is the height (altitude length)of the trapezoid
Perimeter
The perimeter of a trapezoid is simply the sum of all four sides. Since they have no relationship to each other,
there is no formula for it. Simply find the four side lengths and add them up. Use the distance formula to find
any non-horizontal or non-vertical side lengths.
Example
Find the area and perimeter of the trapezoid in the figure above.
Area
1. First, we need the length of the two bases (the parallel sides). These are found by calculating the
distance between the endpoints of the lines segments. Doing this we see that BC = 22 and AD = 47.
2. Then we need the height (altitude). This is the perpendicular distance between the bases. There are
several methods to do this depending on whether the trapezoid is rotated or not. Doing this we see
that the height = 21.
3. Finally we calculate the area using the formula given:
1
b b 
Area  h  b1  b2  OR h  1 2 
2
 2 
1
 22  47 
  21 22  47  OR 21

2
 2 
 725 units2
Things you can do with Coordinate Geometry
Perimeter
1. The perimeter is the sum of the four side lengths. So these are found by calculating the distance
between the endpoints of the lines segments. Doing this we see that BC = 22, AD = 47, AB=22, CD=28.
2. Finally we add them up to get the perimeter: 22 + 22 + 28 + 47 = 119 units
Rotated case
In the previous figure the trapezoid bases are parallel to the x-axis which makes calculations easy. If the figure
is rotated this will not be the case. All the techniques described above will still work, but you have to use the
correct method for finding the distance between two points, and the altitude, which requires the correct
method for finding the perpendicular distance from a point to a line.
Parallelogram
A quadrilateral with both pairs of opposite sides parallel and congruent, and whose location on the coordinate
plane is determined by the coordinates of the four vertices(corners).
From the coordinates of the vertices various properties such as its altitude can be found.
It has all the same properties as a familiar parallelogram:
 Opposite sides are parallel and congruent
 The diagonals bisect each other
 Opposite angles are congruent
Dimensions of a parallelogram
The dimensions of the parallelogram are found by calculating the distance between various corner points.
Recall that we can find the distance between any two points if we know their coordinates. So in the figure
above:
 The height of the parallelogram is the distance between A and B (or C, D).
 The width is the distance between B and C (or A, D).
 The length of a diagonals is the distance between opposite corners, say B and D (or A, C since the
diagonals are congruent).
This method will work even if the parallelogram is rotated on the plane. But if the sides of the
parallelogram are parallel to the x and y-axes, then the calculations can be a little easier.
 The height is the difference in y-coordinates of any top and bottom point - for example A and B.
Things you can do with Coordinate Geometry

The width is the difference in x-coordinates of any left and right point - for example B and D.
Example
 The height of the parallelogram is the distance between the points A and B. (Using C, D will produce
the same result). Since the altitude is vertical we can find this distance by finding the difference of the
y-coordinates: height  26  7  19
 The width is the distance between the points B and C. (Using A, D will produce the same result). Since
the side is horizontal we can find this distance by finding the difference of the x-coordinates:
width  48  18  30
 The length of a diagonals is the distance between B and D. (Using A, C will produce the same result).
Using the formula for the distance between two points, results in:
diagonal 
36 18   26  7 
2
2
 685  26.17