Coordinates
Expected learning outcomes
At the end of this topic, students should be able to:
• state the coordinates of any given point from all four
quadrants of the coordinate plane
• join given coordinate points to draw a polygon
• mark on a coordinate plane the fourth vertex of a
rectangle, square, rhombus or parallelogram
• determine the coordinates of a midpoint of a line
Coordinate Plane
A cartesian plane divides the plane space into two dimensions
and is useful to easily locate the points. It is also referred to as
the coordinate plane. The two axes of the coordinate plane are
the horizontal x-axis and the vertical y-axis. These coordinate
axes divide the plane into four quadrants, and the point of
intersection of these axes is the origin (0, 0). Further, any point in
the coordinate plane is referred to by a point (x, y), where the x
value is the position of the point with reference to the x-axis,
and the y value is the position of the point with reference to the
y-axis.
Coordinates of a Point
A coordinate is an address, which helps to locate a point in space.
For a two-dimensional space, the coordinates of a point are (x, y).
Coordinates are written as (x, y) meaning the point on the x
axis is written first, followed by the point on the y axis.
𝐴 = (4 , 4)
B = (5 , −2)
C = (−2 , 2)
𝐷 = (−3 , −4)
Classwork
STP 7
Page 294
Exercise 16d
Q11
Q12
Join points to draw polygons
A polygon is a closed figure in two dimensions with a
certain number of finite sides. A polygon is drawn in a
coordinate plane by plotting the points and connecting
them. There are a few steps to follow:
• Step 1: Plot each point on the coordinate plane
• Step 2: Label each point of the coordinate plane
• Step 3: Join the adjacent points to form a polygon
Examples
1. Draw a polygon with the following coordinates: A(2, 3),
B(7, 3), C(7, -2), D(2, -2).
2. Draw a polygon with the following points as vertices
The points are: A(-3, 5), B(3, 3), C(6, 5), D(4, -1), E(-1, -2),
F(1, 1).
3. Find the area of the polygon formed by the
coordinates A(1, 3), B(1, -1), and C(6, -1).
Area =10 square units
4. Sam was plotting some random points A(-2, 4), B(-2, -2),
C(6, -2), D(6, 4) on a coordinate plane. Draw the polygon
on a plane and find the perimeter of the polygon.
Perimeter = 28 units
Classwork
STP 7
Page 294
Exercise 16d
Q13
Q14
Q15
Q16
Find the missing vertex of a polygon
1. In the xy-coordinate plane, the coordinates of three
vertices of a parallelogram are (4, 4), (4, 0), and (0, 1).
What are the coordinates of the fourth vertex of the
parallelogram?
Since the left side of the parallelogram must have same
length as the right side (top and bottom sides must also
be equals) so the fourth point must be (0, 5), see the
following figure:
2. The square PQRS has vertices P(1,1), Q(4,1), R(4,4).
Find the coordinates of the fourth vertex S?
3. The rhombus ABCD has vertices A(1,3), B(3,5),
C(3,1). Find the coordinates of the fourth vertex D?
Exercise
STP 7
Page 288
Exercise 16a
Questions 19 to 27
SOLUTIONS
19. T = (4,5), y coordinate is 5
20. P = (4,0), x coordinate is 4
21. S = (1,3), x coordinate is 1
22. R = (0,6), y coordinate is 6
23. Q = (8,5), y coordinate is 5
24. R = (0,6), y coordinate is 0
25. D = (2,5)
26. D = (7,1)
27. D = (4,1)
How To Find The Midpoint
The midpoint of a line segment is a point that lies
exactly halfway between two points. It is the same
distance from each endpoint of the straight line
segment.
Sometimes we can work this out by inspection – this is
easier with positive integer numbers.
For example, given the two points (2,2) and (8,6), the midpoint is
exactly halfway between the two, and lies at (5,4).
We can see that 5 is halfway between 2 and 8, and 4 is halfway
between 2 and 6. Imagining a number line can help with this.
Midpoint of a line formula
If it is not easy to spot the midpoint, or the coordinates
involve fractions or negative numbers, we can use the
midpoint formula.
If the points A(x1,y1) and B(x2,y2) are the endpoints of a
line segment, then the midpoint(M) of the line segment
joining the points A and B is:
𝑥1 + 𝑥2 𝑦1 + 𝑦2
𝑀=( , )
2 2
Examples
1. Find the midpoint of the
line segment joining the
points (0,6) and (4,10).
(2, 8)
2. Find the midpoint of the line segment joining the points
(2,5) and (6,0).
(4, 2.5)
3. Find the midpoint of the line segment joining the points
(-2,7) and (4,6).
(1, 6.5)
4. Three vertices of a rectangle are A(2,-1), B(2,7) and
C(4,7). Plot these points on a graph and hence use it to find
the coordinates of the fourth vertex D. Also, find the
coordinates of the midpoint of CD and the area of the
rectangle?
Exercise
STP 7
Page 295
Exercise 16e
Questions 1, 4, 12, 13, 18, 20 & 21
Answers
Q1: As per your plotting
Q4: As per your plotting
Q12: (4, 2)
Q13: (2,-1)
Q18: (5,7)
Q20: (1.5, 4)
Q21: (-2,2)
Finding a missing endpoint
Sometimes you may be given one endpoint and the midpoint,
and have to work out the other endpoint.
In order to find a missing endpoint when given one endpoint
and the midpoint:
1.Work out how to get from the given endpoint to the
midpoint.
2.Repeat this to get from the midpoint to the missing
endpoint.
3.Write down the coordinates of the missing endpoint.
To get from the first endpoint (1,3) to the midpoint (3,7), we
move 2 in the x-direction and 4 in the y-direction. So we just
repeat this again from the midpoint to find the coordinate of the
other endpoint, which in this case would be (5,11).
Example 1
A line segment joins the points A and B, and has midpoint M.
A has coordinates (4,8) and M has coordinates (6,9).
Find the coordinates of point B.
1. To get from A to M, we add 2 to the x-coordinate of A and
add 1 to the y-coordinate of A.
2. To get from M to B, we add 2 to the x-coordinate of M
and add 1 to the y-coordinate of M.
3. Therefore the coordinates of point B are (8,10).
Example 2
A line segment joins the points A and B, and has
midpoint M.
A has coordinates (−9,4) and M has coordinates (−6,−1).
Find the coordinates of point B.
Step 1. To get from A to M, we add 3 to the x-coordinate
of A and subtract 5 from the y-coordinate of A.
Step 2: To get from M to B, we add 3 to the x-coordinate
of M and subtract 5 from the y-coordinate of M.
Step 3: Therefore the coordinates of point B are (−3,−6).
