GRADIENTS AND LINES, THE EQUATION OF A STRAIGHT LINE

COORDINATE GEOMETRY: GRADIENTS AND LINES, THE EQUATION OF A STRAIGHT LINE

0/1

COORDINATE GEOMETRY (GRADIENTS AND LINES, THE EQUATION OF A STRAIGHT LINE)

LEARNING OBJECTIVES:
By the end of the lesson, students will be able to:
Describe the coordinate of a given point.
Find the gradient of straight line.
Find the distance between 2 points on a straight line.

THE MID-POINT OF A LINE SEGMENT
In general, the coordinates of the mid-point of the line segment joining $(x1, y1) and (x2, y2)$ are given by:
$((x_1+ x_2)/2,(y_1+ y_2)/2)$
$(x_1+ x_2)/2$ – is the mean of x-coordinates
$(y_1+ y_2)/2$ – is the mean of y-coordinates
Example
Find the mid-point of A(4, – 2) and B(- 2, 4)
Answer
Mid-point = $(x_1+ x_2)/2, (y_1+ y_2)/2 = (4+ -2)/2, (-2+ 4)/2=(2/2, 2/2)=(1,1)$

THE GRADIENT/SLOPE
Given that y = mx + c, “m” is the gradient and “C” is the y-intercept.
Also, when there are two points, the gradient is calculated as:
$m= (y_2-y_1)/(x_2-x_1 )$
Examples
1. Given that 2y = 8x – 10, find the gradient and y-intercept
2. Given two points, A(8, – 4) and B(4, 4). Find the gradient of line AB
ANSWERS
1. 2y = 8x – 10 (since y = mx + c)
Divide both sides by 2
$2y/2 = 8x/2 - 10/2$
y = 4x – 5
Hence, m = 4
2. $m= (y_2-y_1)/(x_2-x_1 )=(4--4)/(4-8)=(4+4)/(-4)=-8/4=-2$

THE DISTANCE BETWEEN TWO POINTS $[A(x_(1 ,) y_1 ) and B(x_2,y_2 )$ ]
The distance between two points $A(x_(1 ,) y_1 ) and B(x_2,y_2 )$ , is given as:
$D=\sqrt((x_2-x_1 )^2+(y_2-y_1 )^2 )$
EXAMPLE
What is the distance between two points with coordinates A(2, 3) and B(6, 6)?
ANSWER
$D=\sqrt((x_2-x_1 )^2+(y_2-y_1 )^2 )$
$D=\sqrt((6-2)^2+(6-3)^2 )=\sqrt(4^2+3^2 )=\sqrt(16+9)=\sqrt25=5 units$

THE EQUATION OF A LINE SEGMENT
The equation of a line segment is given as: $y-y_1=m(x-x_1)$
EXAMPLE
Find the equation of the line with gradient 5 passing through the point (–1, –6).
ANSWER
$y-y_1=m(x-x_1)$
y - - 6 = 5(x - - 1)
y + 6 = 5(x + 1)
y = 5x + 5 - 6
y = 5x - 1

ACTIVITY
A. Find the mid-point of the following:
(1) A(2, 4) and B(6, 8). (2) P(–2, 3) and Q(4, 7). (3) M(0, 0) and N(10, 6). (4) A(–4, –2) and B(6, 8).
(5) P(3, –5) and Q(7, 1). (6) M(–6, 4) and N(2, –2). (7) A(1, 9) and B(5, –3). (8) P(–8, –6) and Q(4, 2).
(9) M(10, –4) and N(2, 6). (10) A(–1, –7) and B(5, 3).
B. Find the gradient of the following:
(11) (2, 3) and (6, 11). (12) (–4, 1) and (2, 7) . (13) (0, 5) and (4, 5). (14) (3, –2) and (3, 6).
(15) (–2, –4) and (4, 8). (16) (1, 7) and (5, 3). (17) (–6, 2) and (0, –4). (18) (2, –1) and (8, 5).
(19) (–3, 6) and (1, –2) (20) (4, 9) and (10, 3).
C. Find the distance between the following points:
(21) (2, 3) and (6, 7). (22) (–1, –2) and (3, 2). (23) (0, 0) and (5, 12). (24) (–4, 3) and (2, –3).
(25) (1, 5) and (4, 9). (26) (–2, –6) and (–2, 2). (27) (3, –4) and (7, –4). (28) (–5, –1) and (1, 7).
(29) (2, 1) and (6, –3). (30) (–7, 4) and (–1, –2)

D. (31) Find the equation of the line with gradient 2 passing through (1, 3).
(32) Find the equation of the line with gradient –1 passing through (4, 6).
(33) Find the equation of the line passing through the points (2, 1) and (6, 9).
(34) Find the equation of the line passing through the points (–3, 5) and (3, –1).
(35) Find the equation of the line with gradient 0 passing through (–2, 4).
(36) Find the equation of the line passing through (3, –2) and having gradient 4.
(37) Find the equation of the line passing through the points (1, –3) and (5, 1).
(38) Find the equation of the line with gradient –3 passing through the point (–2, 1).
(39) Find the equation of the line passing through (0, –4) and (6, 2).
(40) Find the equation of the line with gradient ½ passing through the point (2, –1).

FINAL ANSWERS:
(1) (4, 6) (2) (1, 5) (3) (5, 3) (4) (1, 3) (5) (5, –2) (6) (–2, 1) (7) (3, 3)
(8) (–2, –2) (9) (6, 1) (10) (2, –2) (11) 2 (12) 1 (13) 0 (14) Undefined
(15) 2 (16) –1 (17) –1 (18) 1 (19) –2 (20) –1 (21) $4\sqrt2$
(22) $4\sqrt2$ (23) 13 (24) $6\sqrt2$ (25) 5 (26) 8 (27) 4 (28) 10
(29) $4\sqrt2$ (30) $6\sqrt2$ (31) y = 2x + 1 (32) 𝑦 = −𝑥 + 10 (33) y = 2x − 3 (34) y = − x + 2
(35) y = 4 (36) y = 4x − 14 (37) y = x − 4 (38) y = − 3x − 5 (39) y = x − 4 (40) $y=1/2 x-4$

6
2
6
2