Coordinate geometry

1. Coordinate
Geometry
The Cartesian Plane and Gradient ;

2. Basic Terminology
 The figure on the right
shows 2 perpendicular
lines intersecting at the
point O. This is called
the Cartesian Plane.
 O is also called the
origin.
 The horizontal line is
called the x-axis and
the vertical line is
called the y-axis

3. Coordinates of a Point
 The position of any
point in the Cartesian
Plane can be determined
by its distance from each
axes.
 Example: Point A is 3
units to the right of the y-
axis and 1 unit above the
x-axis, its position is
described by the
coordinate (3, 1).
 Similarly, the coordinates
of Points B, C and D are
determined as shown.

4. Question
 What
coordinate
represents the
origin O ?
 Ans:

5. Summary
 Any point, P, in the plane can be located by it’s
coordinates (x, y).
 We call x the x - coordinate or abscissa of P
and y the y - coordinate or ordinate of P.
 Hence, we say that P has coordinates (x, y).

6. Write the coordinates of the following points 1
P
S
Q
R

7. Gradient (or slope)
 The steepness of a line
is called its GRADIENT
(or slope).
 The gradient of a line
is defined as the ratio
of its vertical distance
to its horizontal
distance.
l
vertical distance
gradient
horizontal distance
=

8. Examples of Gradient
What is the gradient of the driveway?
2
17
vertical distance
gradient
horizontal distance
= =Ans:
Note: Gradient has no units!

9. Examples of Gradient
An assembly line is pictured below. What is the
gradient of the sloping section?
0.85 17
15 300
vertical distance
gradient
horizontal distance
= = =Ans:

10. Examples of Gradient
The bottom of the playground slide is 2.5 m from the foot of
the ladder. The gradient of the line which represents the slide
is 0.68. How tall is the slide?
0.68
2.5
1.7
x
x m
=
=
Ans:

11. Question
 For safety considerations, wheelchair ramps are
constructed under regulated specifications. One regulation
requires that the maximum gradient of a ramp exceeding
1200 mm in length is to be
(a) Does a ramp 25 cm high with a horizontal length of
210 cm meet the requirements?
(b) Does a ramp with gradient meet the specifications?
(c) A 16 cm high ramp needs to be built. Find the minimum
horizontal length of the ramp required to meet the
specifications.
1
14
1
18
Ans: No
Ans: Yes
Ans: 224 cm

12. Horizontal and Vertical Lines
 The gradient of a horizontal line is ZERO
(Horizontal line is flat – No Slope)
 The gradient of a vertical line is INIFINITY
(Vertical line – gradient is maximum)

13. Finding the gradient of a straight
line in a Cartesian Plane
(a) Positive Gradients
 Lines that climb from left to the right are said
to have positive gradient/slope:
(b) Negative Gradients
 Lines that descend from left to the right are
said to have negative gradient/slope:

14. Examples
10
1.5
15
gradient = =
8
0.5
16
gradient = =
(0, 10)
(-15, 0)
Write down the coordinates of the points given
(0, -8)
(16, 0)

15. Examples
6
2
3
gradient = − = −
12
3
4
gradient = − = −
(0, 6)
(3, 0)
(-4, 0)
(0, -12)

16. Summary
Infinite

17. Gradient Formula
 So far, we have determined the gradient using the
idea of
 Using the above, we must always remember to add
a negative sign to slopes with negative gradient.
 Now, let’s look at the formula to determine gradient.
The formula will take into consideration the sign of
the slope
vertical distance
gradient
horizontal distance
=

18. Gradient Formula
2 1
2 1
gradient
vertical distance
horizontal distance
y y
x x
−
=
−
=
A(x1,y1)
x1 x2
y1
y2
B(x2,y2)
Horizontal = x2 – x1
Vertical = y2 – y1
y
x

19. How to apply gradient
formula
 Write down the coordinates of 2 points
on the line: (x1, y1) and (x2, y2)
 If the coordinate is negative, include
its sign
 Apply the formula

20. Examples:
L1:
2 points on the line
are (1, 4) and (0, 1)
Tip: Choose points
that are easy to
read!
2 1
2 1
1 4
0 1
3
1
3
y y
gradien
x x
t
−
−
=
−
=
−
−
=
−
=
(1, 4)
(0, 1)
1 square represents 1 unit on both axes

21. Examples:
L2:
2 points on the line
are (1, 1) and (3, 3)
Tip: Choose points
that are easy to
read!
2 1
2 1
3 1
3 1
2
2
1
y y
x x
gradient =
−
−
=
−
=
=
−
(3, 3)
(1, 1)
1 square represents 1 unit on both axes

22. Examples:
L3:
2 points on the line
are (3, 1) and (1, 0)
2 1
2 1
1 0
3 1
1
2
y y
x
gradient
x
=
−
−
−
=
=
−
(3, 1)
(1, 0)
1 square represents 1 unit on both axes

23. Examples:
L4:
2 points on the line
are (3, -1) and (-3, -3)
2 1
2 1
3 ( 1)
3 3
2
6
1
3
gra
y y
x x
dient =
− − −
=
− −
−
=
−
=
−
−
(3, -1)
(-3, -3)
1 square represents 1 unit on both axes

24. Examples:
L5:
2 points on the line
are (0, 1) and (1, -2)
2 1
2 1
2 1
1 0
3
y y
grad
x
i
x
ent
−
−
=
− −
=
−
= −
(0, 1)
(1, -2)
1 square represents 1 unit on both axes

25. Examples:
L6:
2 points on the line
are (0, 0) and (-4, 4)
2 1
2 1
4 0
4 0
1
y y
grad
x
i
x
ent
−
−
=
−
=
− −
= −
(-4, 4)
(0, 0)
1 square represents 1 unit on both axes

26. Examples:
L7:
2 points on the line
are (4, -2) and (-2, 2)
2 1
2 1
2 2
4 ( 2)
4
6
2
3
y y
x
gradien
x
t =
− −
=
− −
−
=
= −
−
−
(-2, 2)
(4, -2)
1 square represents 1 unit on both axes

27. Examples:
L8:
2 points on the line
are (0, -2) and (-3, -1)
2 1
2 1
2 ( 1)
0 ( 3)
1
3
grad
y y
i
x x
ent =
− − −
=
− −
= −
−
− (-3, -1)
(0, -2)
1 square represents 1 unit on both axes

28. Question
 Is there a difference between
2 1 1 2
2 1 1 2
?
y y y y
and
x x x x
− −
− −
 Is there a difference between
2 1 2 1
2 1 2 1
?
y y x x
and
x x y y
− −
− −
Ans: No.
1 2 2 1
1 2 2 1
( )
( )
y y y y
x x x x
− − −
=
− − −
Ans: Yes!
2 1
2 1
horizontal distance
gradient
vertical distance
x x
y y
−
= ≠
−

29. Solution to Exercise 2
In order from smallest to largest gradient: e, b, a, d, c

30. 2gradient = 5gradient =
1
4
gradient =

31. 1
3
gradient = − 4gradient = − 1gradient = −

32. 1
2
gradient = 3gradient =
3
2
gradient =

33. 1gradient = −
3
4
gradient = −
1
2
gradient = −

34. 1
2
gradient =
7
8
gradient = −
3
Horizontal Line: Zero
Vertical Line: Inifinity

35. Thank…..
You…….