Week 4 – SS1 Third Term Further Mathematics Notes

 WEEK 4
Topic:         Straight line
Sub-topic:
     Angle of slope and angle between lines
Duration:     80 minutes
Learning Objectives: By the end of the lesson, students should be able to calculate the angle of slope and angle between two lines.
Reference Materials: New Further Mathematics Project 2 by M. R Tuttuh Adegun
Previous Knowledge: Students can draw the graph of a linear equation (straight-line graph).
Instructional Materials: Graph board and graph book.
Content:                    ANGLE OF SLOPE
Example:    Find the gradient of the line joining (3, 2) and (7, 10) and the angle of slope of the line.
Solution
Let m be the gradient of the line, then
        m =
Let be the angle of slope of the line; then:
        
        

 ANGLE BETWEEN TWO LINES
Condition for Parallelism
If two lines are parallel, the angle between them is zero, hence
Example:    Determine if AB is parallel to PQ in each of the following.

1. A(3, 1); B(4, 3) and P(4,6); Q(5, 8)
2. A(-1, -2); B(2, -3) and P(5, 4) ; Q(6, 7)

Solution

1. Let be the gradient joining A and B and be the gradient joining P and Q.

    

    Since ; AB||PQ
   

1. Let be the gradient joining A and B and be the gradient joining P and Q.

    

    Since ; AB is not parallel to PQ
   
   

CONDITION FOR PERPENDICULARITY
    If the lines are perpendicular, and ; therefore:
            1 +
                
                
Example:    Determine if AB is parallel to PQ in each of the following.

1. A(5, -1); B(3, 2) and P(2, 4); Q(5, 6)
2. A(-1, -2); B(2, -3) and P(5, 4) ; Q(6, 7)

Solution

1. Let be the gradient joining A and B and be the gradient joining P and Q.

    

    Since ; AB is perpendicular to PQ
   

1. Let be the gradient joining A and B and be the gradient joining P and Q.

    

    Since ; AB is perpendicular to PQ
   
   

EQUATION OF A LINE
The equation of a straight line is given by:        y =mx + c
Example:    Find the gradient and intercept on the y-axis of the following lines:

1. y = 3x – 4
2. y = – ½x – 3

Solution:

1. Compare y = 3x – 4 with y = mx + c ; Hence the gradient is 3, intercept on y-axis is -4
2. Gradient is – ½ , intercept on y-axis

GRADIENT AND ONE POINT FORM
Example:    Find the equation of a straight line of slope 2, if it passes through the point (3, -2)
                    y –
                m = 2;
        Hence the equation of the straight line is:
            y – (-2) = 2(x – 3)
            y + 2 = 2x – 6
            y = 2x -6 -2 = 2x – 8
                y = 2x – 8