CHAPTER 5 The Straight Line

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.

Equation of straight line The Straight Line Gradient Intercept y-intercept x-intercept Equation of straight line Gradient = m y-intercept = c Parellel lines

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)

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

(B)Determine the ratio of the vertical distance to the horizontal distance Let us look at the ratio of the vertical distance to the horizontal distances of the slope as shown in figure.

Therefore, Solution: Vertical distance = 10 m Horizontal distance = 16 m Therefore, Solution:

5.2 GRADIENT OF THE STRAIGHT LINE IN CARTESIAN COORDINATES Coordinate T = (X2,Y1) horizontal distance = PT = Difference in x-coordinates = x2 – x1 Vertical distance = RT = Difference in y-coordinates = y2 – y1 y R(x2,y2) y2 – y1 x2 – x1 T(x2,y1) P(x1,y1) x

Solution: REMEMBER!!! For a line passing through two points (x1,y1) and (x2,y2), where m is the gradient of a straight line

Example 2 Determine the gradient of the straight line passing through the following pairs of points P(0,7) , Q(6,10) L(6,1) , N(9,7) Solution:

(C) Determine the relationship between the value of the gradient and the Steepness Direction of inclination of a straight line What does gradient represents?? Steepness of a line with respect to the x-axis.

B a right-angled triangle. Line AB is a slope, making an angle with the horizontal line AC A C

When gradient of AB is positive: When gradient of AB is negative: y y B B x x A A inclined upwards acute angle is positive inclined downwards obtuse angle. is negative

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 MN PQ RS UV V(1,4) N(3,3) Q(-2,4) S(-3,1) x M(-2,-2) R(3,-1) P(2,-4) U(-1,-4)

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

Lines Gradient AB y A B x

Lines Gradient CD Undefined y D C x

Lines Gradient EF Positive y F E x

Lines Gradient GH Negative y H G x

AB CD EF GH Undefined Positive Negative Lines Gradient y D H F A B G E CD Undefined EF Positive GH Negative y D H F A B G E C x

5.3 Intercepts Another way finding m, the gradient: y-intercept x-intercept Another way finding m, the gradient:

5.4 Equation of a straight line Slope intercept form y = mx + c Point-slope form given 1 point and gradient: given 2 point:

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 y=2x-6 gradient is 2. 2y=4x+3 gradient is 2. Since their gradient is same hence they are parallel.