# Lesson Notes By Weeks and Term - Senior Secondary School 3

Coordinate Geometry of straight lines

SUBJECT: MATHEMATICS

CLASS:  SS 3

DATE:

TERM: 2nd TERM

REFERENCE TEXT

• New General Mathematics for SS book 3 by J.B Channon
• Essential Mathematics for SS book 3
• Mathematics Exam Focus
• Waec and Jamb past Questions

WEEK THREE

TOPIC: Coordinate Geometry of straight lines

• Coordinate Geometry of straight lines:
• Gradient and Intercepts of a line
• Angle between two intersecting straight lines and application

Gradient and Intercepts of a line

Gradient of a line of the form y = mx + c, is the coefficient of x, which is represented by m and c is the intercept on the y axis.

Example

1. Find the equation of the line with gradient 4 and y-intercept -7.

Solution

m = 4, c = - 7,

Hence, the equation is; y =4x - 7.

Evaluation:

1.What is the gradient and y intercept of the line equation 3x -5y +10=0 ?

1. Find the equation of the line with gradient - 9 and y-intercept 4.

The equation of the line can be calculated given one point (x, y) and gradient (m) by using the formula; y - y1= m(x - x1)

Example

Find the equation of the line with gradient -8 and point(3, 7).

Solution

m = - 8, (x1, y1) =(3,7)

Equation: y - 7 = - 8(x - 3)

y = -8x + 24 +7

y = -8x + 31

Evaluation:

1. Find the equation of the line with gradient 5 and point(-2, -7).
2. Find the equation of the line with gradient -12and point (3, -5).

Two Point Form:

Given two points (x1, y1) and (x2, y2), the equation can be obtained using the formula:

y2 - y1 = y - y1

x2 - x1      x - x1

Example: Find the equation of the line passing through (2,-5) and (3,6).

Solution

6 - (-5)/3 - 2 = y - (-5)/x - 2

11 = y + 5/x - 2

11(x - 2) = y + 5

11x - 22 = y + 5

y - 11x + 27 = 0

Evaluation:

1.Find the equation of the line passing through (3, 4) and (-1, -2).

2.Find the equation of the line passing through (-8, 5) and (-6, 2).

Angles between Lines

Parallel lines:

The angle between parallel lines is 00 because they have the same gradient

Perpendicular Lines:

Angle between two perpendicular lines is 900 and the product of their gradients is – 1. Hence, m1m2 = - 1

Examples:

1. Show that the lines y = -3x + 2 and y + 3x = 7 are parallel.

solution:

Equation 1: y = -3x + 2,   m1 = -3

Equation 2:  y + 3x = 7,

y = -3x + 7, m2 = - 3

since; m1 = m2 = - 3, then the lines are parallel

1. Given the line equations x = 3y + 5 and y + 3x = 2, show that the lines are perpendicular.

solutions:

Equation 1:     x = 3y + 5,   make y the subject of the equation.

3y = x + 5

y = x/3 + 5/3

m1 = 1/3

Equation 2:    y + 3x = 2,

y = - 3x + 2,   m2 = -3

hence: m1 x m2 = 1/3 x – 3 = - 1

since: m1m2 = - 1, then the lines are perpendicular.

Evaluation: State which of the following pairs of lines are: (i) perpendicular   (ii) parallel

(1)    y = x + 5 and y = - x + 5      (2). 2y – 6 = 5x and 3 – 5y = 2x    (3) y = 2x – 1 and 2y – 4x = 8

Angles between Intersecting Lines:

y

y = mx + c

θ                                                x

0

The gradient of y = mx + c is tan θ.    Hence m = tan θcan be used to calculate angles between two intersecting lines. Generally the angle between two lines can be obtained using: tan 0 = m2 -m1

1 + m1m2

Example: Calculate the acute angle between the lines y=4x -7 and y = x/2 + 0.5.

Solution:

Y=4x -7, m1= 4, y=x/2+0.2, m2 =1/2.

Tan O= 0.5 - 4.       = -3.5/3

1 + (0.5*4)

Tan O =- 1.1667

O=tan-1(-1.1667) = 49.4

Evaluation:Calculate the acute angle between the lines y=3x -4 and x - 4y +8 = 0.

General Evaluation:

1.Calculate the acute angle between the lines y=2x -1 and  2y + x = 2.

2.If the lines 3y=4x -1 and qy= x + 3 are parallel to each other, find the value of q.

3.Find the equation of the line passing through (2,-1) and gradient 3.

Reading Assignment: NGM for SS 3 Chapter 9 page 75 – 81

Weekend Assignment

1.Find the equation of the line passing through (5,0) and gradient 3.

2.Find the equation of the line passing through (2,-1) and (1, -2).

1. Two lines y=3x - 4 and x - 4y + 8=0 are drawn on the same axes.

(a) Find the gradients and intercepts on the axes of each line.

(b) Find the equation parralel to x -4y + 8=0 at the point (3, -5)