Olympic College - Topic 6 Cartesian Coordinates and Slopes
Topic 6 Cartesian Coordinates and Slopes
1. Cartesian Coordinate System
Definition: The “Cartesian Coordinate System” way of representing the position of a point in
two dimensional space. It is constructed with two perpendicular axes the horizontal one is called
the x-axis while the vertical one is called the y-axis. The two axes cross at the origin and we give
the location of any point in terms of a number pair (called coordinates) written in the form (a,b)
where a is the distance along the x-axis of the point and b is the distance along the y-axis of the
point. By this means every point in two dimensional space can be represented by a unique
coordinate.
Example 1: Plot the points with coordinates A(4,3) , B (-2,4) , C(0,-3)
Solution:
y
B(-2,4)
5
4
A(4,3)
3
4 up
2
3 up
1 4 to the right
-5 -4 -3 -2 -1 0
-1
1
2
3
x
4
5
y
y
5
5
4
4
3
3
2
2
1
1
2 to the left
-5 -4 -3 -2 -1 0
-1
1
x
2
3
4
0 to the right
-5 -4 -3 -2 -1 0
-1
5
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
1
2
C(0,-3)
Example 2: What are the coordinates of the points on the Cartesian graph below.
Solution:
A(-2,4)
y
A
B(2,4)
5
B
4
3
C(-3,0)
D
2
D(4,2)
E(-3,-3)
F(0,-4)
G(3,-3)
C
1
-5 -4 -3 -2 -1 0
-1
E
-2
-3
x
1
2
3
4
5
G
F- 4
-5
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x
4
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
Exercise 1A:
1.
Plot the points with coordinates A(5,1) , B (-3,2) , C(0,4), D(-2,-4) , E (-2,0) , F(4,3).
2.
(a) What are the coordinates of the following points?
y
(b) Which points have an x-coordinate of – 3?
5
A
B
4
(c) Which points have a y-coordinate of – 3?
3
(d) Which points have negative x-coordinates?
2
1
C
(e) Which points have positive y-coordinates?
-5
(f) Which point(s) have both their x-coordinate
and y-coordinate equal to each other?
-4
D
x
-3
-2
-1
0
1
2
3
4
-1
-2
E
(g) Which point(s) have both their x-coordinate
and y-coordinates negative?
G
-3
F
-4
(h) Which point(s) have a y-coordinate that is
two more than its x-coordinate?
-5
(i) Which point(s) have a y-coordinate that is two less than its x-coordinate?
(k) Which point(s) have a y-coordinate that is four less than its x-coordinate?
(l) Which point(s) have a y-coordinate that is half its x-coordinate?
3.
On the coordinate plane below, which point represents the coordinates (4,-2)
y
A
5
4
3
C
2
1
-5 -4 -3 -2 -1 0
-1
x
1
2
-2
-5
4
5
B
-3
-4
3
D
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
2. Slope of a Line Segment
The slope of a line segment is a number that measures how steep a line is. It is defined as the
ratio between the vertical distance called the “Rise” and the horizontal distance called the “Run”.
A graphical representation of a line segment with a positive slope is given below.
y
Run = 6
5
4
Slope =
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
Slope =
8
6
Slope =
4
3
3
2
Rise = 8
1
x
-5 -4 -3 -2 -1 0
-1
1
2
3
4
5
-2
-3
-4
-5
The other possible slopes of line segments are negative slopes, zero slopes and undefined slopes.
An example of each type of slope is given below.
Notice we typically use the letter m to represent the slope of a line segment.
Example 1:
y
5
Rise = 0
4
m
=
4
3
Run = 6
2
1
x
1
2
3
4
5 -5 -4 -3 -2 -1 0
-1
x
1
2
3
4
5
Rise = 8
1
-5 -4 -3 -2 -1 0
-1
-2
-2
-2
-3
-3
-3
Run = 6
-4
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
−8
6
4
−3
x
1
2
3
-4
Run = 0
-5
Negative Slope:
=
4
2
-5
Slope
5
3
-4
=
5
2
-5 -4 -3 -2 -1 0
-1
Slope
y
3
1
Rise = - 8
y
-5
Zero Slope (horizontal line)
Slope
=
Slope
=
m
=
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
0
6
0
Undefined Slope(vertical line)
Slope
=
Slope
=
m
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
8
0
= undefined
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
It is easy to see when a line segment will have a positive, negative, zero or undefined slope by
just looking at the graph.
y
Example 2: (a) Which of the following line segments
have positive slopes?
1
2
(b) Which of the following line segments
have negative slopes?
3
8
6
(c) Which of the following line segments
have a slope of zero?
(d) Which of the following line segments
have undefined slopes?
Solution: (a) Lines which go up (increase) as
you move from left to right will
have positive slopes.
So lines 1,2 & 5 are positive slopes.
4
5
7
(b) Lines which go down (decrease)
as you move from right to left
have negative slopes.
So lines 3,4 & 6 have negative slopes.
y
y
1
x
2
3
6
x
x
4
5
Solution: (c) Lines which are horizontal will have a slope of zero.
So line 8 has a slope of zero.
Solution: (d) Lines which are vertical will have an undefined slope.
So line 7 will have an undefined slope.
y
8
x
7
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
We can also find the slope of a line segment by using the coordinates of the ends of the line.
The slope of the line segment that joins the points A(x1,y1) and B(x2,y2) is given by the formula.
y
Run = x2 – x1
B(x2,y2)
Slope =
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
x
Rise = y2 – y1
m
=
𝑦2− 𝑦1
𝑥2− 𝑥1
A(x1,y1)
We only need the coordinates of the end points of the line segment in order to calculate its slope.
This is useful as we do not need to plot the point or be given a Cartesian Graph.
Example 3: (a) Find the slope of the line segment joining the points A(3,4) and B(7,9).
(b) Find the slope of the line segment joining the points C(-7,2) and D(3,– 4).
(c) Find the slope of the line segment joining the points E(2,5) and F(7,5).
(d) Find the slope of the line segment joining the points G(2,-2) and H(2,9).
Solution:
(a) The points are A(3,4) and B(7,9) so x1 = 3 , y1 = 4, x2 = 7 and y2 = 10
Slope
=
m
𝑦
𝑦
= 𝑥2−𝑥1 =
2− 1
10−4
=
7−3
6
2
= 3
(b) The points are C(-7,2) and D(3,– 4) so x1 = - 7 , y1 = 2, x2 = 3 and y2 = - 4
Slope
=
m
𝑦
𝑦
= 𝑥2−𝑥1 =
2− 1
−4−2
3−(−7)
=
−6
10
3
= −5
(c) The points are E(2,5) and F(7,5) so x1 = 2 , y1 = 5, x2 = 7 and y2 = 5
Slope
=
m
𝑦
𝑦
= 𝑥2−𝑥1 =
2− 1
5−5
7−2
=
0
5
= 0 (A horizontal line)
(d) The points are G(2,-2) and H(2,9) so x1 = 2 , y1 = – 2 , x2 = 2 and y2 = 7
Slope
=
m
𝑦
𝑦
= 𝑥2−𝑥1 =
2− 1
7−(−2)
2−2
=
9
0
= undefined
(A vertical line)
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
There are a number of useful facts that use slopes. Two of the most common are that if the slope
of two line segments is equal then the line segments are parallel. The other fact is that if two line
segments are perpendicular (cross at 900) then the product of their slopes will be – 1. This last
fact only works if neither of the lines is horizontal or vertical.
Example 4: Which of the following pairs of line segments are parallel and which are
perpendicular.
y
y
A
E
H
D
x
x
F
G
B
Solution:
C
First find the coordinates of the end points of the line segments.
A(2,4) and B(-4,-4) so x1 = 2 , y1 = 4, x2 = -2 and y2 = -4
𝑦
𝑦
−4−4
−8
Slope = mAB = 𝑥2−𝑥1 = −4−2 =
=
−6
2− 1
4
3
C(-1,-4) and D(2,0) so x1 = -1 , y1 = -4, x2 = 2 and y2 = 0
𝑦
𝑦
0−(−4)
Slope = mCD = 𝑥2−𝑥1 = 2−(−1) =
2− 1
4
3
Since the slopes of the two line segments are equal we can conclude that the two
line segments are parallel.
Solution:
First find the coordinates of the end points of the line segments.
E(-4,3) and F(4,-1) so x1 = -3 , y1 = 3, x2 = 4 and y2 = -1
𝑦
𝑦
Slope = mEF = 𝑥2−𝑥1
2− 1
−1−3
= 4−(−4) =
4
1
−8 = −2
G(1,-2) and H(3,2) so x1 = 1 , y1 = -2, x2 = 3 and y2 = 2
𝑦
𝑦
Slope = mCD = 𝑥2−𝑥1 =
2− 1
2−(−2)
3−1
=
4
2
=2
1
Conclusion: Since the produce of the two slopes mEF * mCD = − 2 ∙ 2 = −1 we
can conclude that the two line segments are perpendicular.
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
Exercise 2A:
1.
Calculate the Slope of each line below, leaving your answer as a fraction in its simplest
form where necessary.
y
2
1
x
3
5
4
2.(a)
2.(b)
y
Which of the following line segments
have positive slopes?
1
2
Which of the following line segments
have negative slopes?
3
6
2.(c)
Which of the following line segments
have a slope of zero?
8
x
7
5
4
2.(d)
Which of the following line segments
have undefined slopes?
3.(a)
Find the slope of the line segment joining the points A(2,7) and B(4,15).
3.(b)
Find the slope of the line segment joining the points C(–2,8) and D(2,– 4).
3.(c)
Find the slope of the line segment joining the points E(–5, –2) and F(3,10).
3.(d)
Find the slope of the line segment joining the points G(8, –1) and H(8,4).
3.(e)
Find the slope of the line segment joining the points J(–3,5) and K(3,5).
3.(f)
Find the slope of the line segment joining the points L(–5, –5) and M(15,– 10).
3.(g)
Find the slope of the line segment joining the points N(1.2,0) and O(–0.8,0.5).
3.(h)
Find the slope of the line segment joining the points P( 2 , 3 ) and Q( 4 , 5 )
1
1
1 1
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
4. Which of the following pairs of line segments are parallel and which are perpendicular or
neither?
y
y
y
E
A
C
H
L
x
x
x
K
J
F
D
G
I
B
5. Which of the following pairs of line segments are parallel and which are perpendicular or
neither?
y
y
y
A
C
L
E
H
x
B
D
J
x
G
F
x
K
I
6. Is the line segment joining the points A(3,5) and B(7,7) parallel or perpendicular to the line
segment joining the points C(– 1 ,2) and D(1,– 2)?
7. Is the line segment joining the points A(2,2) and B(0,4) parallel or perpendicular to the line
segment joining the points C(– 3 ,0) and D(– 7 ,4)?
8. Is the line segment joining the points A(3,5) and B(3,7) parallel or perpendicular to the line
segment joining the points C(– 1 ,2) and D(7,2)?
9. Is the line segment joining the points A(13,– 8 ) and B(3, – 3 ) parallel or perpendicular to
the line segment joining the points C(– 4 ,3) and D(– 14 ,8)?
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Olympic College - Topic 6 Cartesian Coordinates and Slopes
Solutions.
y
Exercise 1A:
5
1.
2.(a)
C
4
F
3
B
2
1
A x
E
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
G
-3
2.(b)
2.(c)
2.(d)
2.(e)
2.(f)
2.(g)
2.(h)
2.(i)
2.(k)
2.(l)
A(-2,4) B(2,4) C(-3,0) D(4,2)
E(-3,-3) F(0,-4) G(3,-3)
C and E
E and G
A,C and E
A,B and D
E
E
B
D
F
D
3.
B
D
-4
-5
Exercise 2A:
1.
2
m1 = 3
m2 = 0
1
m3 = −1
m4 = 7
m5 = undefined
2.(a)
2.(c)
Lines 2,4 and 5
Line 3
2.(b)
2.(d)
Lines 1,7 and 8
Line 6
3.(a)
3.(c)
Slope = mAB = 4
3
Slope = mEF = 2
3.(b)
3.(d)
Slope = mCD = – 3
Slope = mGH undefined
3.(e)
Slope = mJK = 0
3.(f)
Slope = mLM = − 2
3.(g)
Slope = mNO = – 0.25
3.(h)
Slope = mPQ = 15
1
8
4.
AB and CD are perpendicular; EF and GH neither; IJ and KL are parallel.
5.
AB and CD are neither; EF and GH perpendicular; IJ and KL are parallel.
6.
perpendicular
7.
parallel
8.
perpendicular
9.
parallel
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