2. Learning Objectives
5.1 Understand the concept of gradient of a
straight line.
5.2 Understand the concept of gradient of a
straight line in Cartesian coordinates.
5.3 Understand the concept of intercept.
5.4 Understand and use equation of a straight
line.
5.5 Understand and use the concept of parallel
lines.
4. 5.1 graDient OF a
straigHt Line
(A) Determine the vertical and horizontal distances
between two given points on a straight line
F
E G
Example of application: AN ESCALATOR.
EG - horizontal distance(how far a person goes)
GF - vertical distances(height changed)
5. Example 1
State the horizontal and vertical
distances for the following case.
10 m
16 m
Solution:
The horizontal distance = 16 m
The vertical distance = 10 m
6. (B)Determine the ratio of the vertical
distance to the horizontal distance
10 m
16 m
Let us look at the ratio of the vertical distance
to the horizontal distances of the slope as
shown in figure.
8. 5.2 GRADIENT OF THE STRAIGHT LINE IN
CARTESIAN COORDINATES
y • Coordinate T = (X2,Y1)
• horizontal distance
R(x2,y2)
= PT
= Difference in x-coordinates
y 2 – y1
= x2 – x 1
x2 – x 1 • Vertical distance
P(x1,y1) T(x2,y1)
= RT
x = Difference in y-coordinates
0
= y2 – y 1
9. Solution:
vertical distance
gradient of PR =
horizontal distance
RT
=
PT
y 2 − y1
=
x2 − x1
REMEMBER!!!
For a line passing through two points (x1,y1) and (x2,y2),
y2 − y1
m=
x2 − x1
where m is the gradient of a straight line
10. Example 2
• Determine the gradient of the straight line
passing through the following pairs of points
i) P(0,7) , Q(6,10)
ii)L(6,1) , N(9,7)
Solution:
10 − 7 7 −1
Gradient PQ = Gradient LN =
6−0 9−6
3 units 6 units
= =
6 units 3 units
1 =2
=
2
11. (C) Determine the relationship between
the value of the gradient and the
(i)Steepness
(ii)Direction of inclination of a straight line
• What does gradient represents??
Steepness of a line with respect to the x-
axis.
12. B
• a right-angled triangle. Line
AB is a slope, making an
angle θ with the horizontal
line AC
θ
A C
vertical distance
tan θ =
horizontal distance
= gradient of AB
13. y y
B
B
θ θ
x x
0 0 A
A
When gradient of AB is When gradient of AB is
positive: negative:
• inclined upwards • inclined downwards
• acute angle • obtuse angle.
• tan θ is positive • tan θ is negative
14. Activity:
Determine the gradient of the given lines in figure
and measure the angle between the line and the x-
axis (measured in anti-clocwise direction)
y
Line Gradient Sign θ
V(1,4) N(3,3)
Q(-2,4)
MN
S(-3,1)
0
x PQ
M(-2,-2)
R(3,-1) RS
U(-1,-4)
P(2,-4)
UV
15. REMEMBER!!!
The value of the gradient of a line:
• Increases as the steepness increases
• Is positive if it makes an acute angle
• Is negative if it makes an obtuse angle
20. Lines Gradient
y
AB
D
H
F
0
A B
G
CD Undefined
E
EF
C
Positive
0 x
GH Negative
21. 5.3 Intercepts
y-intercept
x-intercept
• Another way finding m, the gradient:
y - intercept
m=−
x - intercept
22. 5.4 Equation of a straight line
• Slope intercept form
y = mx + c
• Point-slope form
given 1 point and gradient:
y − y1 = m( x − x1 )
given 2 point:
y − y1 y 2 − y1
=
x − x1 x2 − x1
23. 5.5 Parallel lines
• When the gradient of two straight lines
are equal, it can be concluded that the
two straight lines are parallel.
Example:
Is the line 2x-y=6 parallel to line 2y=4x+3?
Solution:
2x-y=6y→ → is 2.
y=2x-6 gradient
3
2y=4x+3 → + 2 →
y = 2x gradient is 2.
Since their gradient is same hence they are parallel.