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Brainline Mathematics Gr.12 Mod2-Unit AG 1 Revise Analytical Geometry P.1
Module: 2 From Grade 10:
Analytical geometry: • Distance formula:
Unit: AG1 The distance between A(x , y ) and B(x , y ) is:
A A B B
22
Analytical Geometry: AB = (-x xy)+(-y)
Revision AB AB
or
AB2 = 22
(-xx)+(y-y)
Assessment standards: AB AB
10.3.3 • Mid-point formula:
Represent geometric figures on a
Cartesian coordinate system, and The coordinates of the midpoint of AB is
derive and apply, for any two
x , y ) and (x , y ), a
points ( 1 1 2 2 x + xy+y
ABAB
formula for calculating: M ;
22
(a) the distance between the two
points;
(b) the gradient of the line segment The uppercase letter M is used to denote the
points; midpoint.
joining the
(c) the coordinates of the mid-point
of the line segment joining the • The gradient of a straight line:
points.
11.3.3 Use a Cartesian coordinate The gradient of the straight line passing through
system to derive and apply:
(a) the equation of a line through A(xy; ) and B(x; y) is
two given points; AA BB
(b) the equation of a line through
one point and parallel or yyyy
AB BA
perpendicular to a given line; m = or
AB x xxx
(c) the inclination of a line.
AB BA
The lower case letter m is used to denote the
gradient.
From Grade 11:
• The equation of a straight line:
can be written in one of the following forms:
y = mx + c m is the gradient
c is the y-intercept
(gradient-intercept formula)
y – y = m(x – x ) line with gradient m which passes
1 1
through (x ; y )
1 1
(point-gradient formula)
a is the x-intercept and
x y
+ = 1
b is the y-intercept
ab (dual intercept formula)
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Brainline Mathematics Gr.12 Mod2-Unit AG 1 Revise Analytical Geometry P.2
x = number: This is a vertical line parallel to the y - axis.
The gradient is undefined.
(x = 0) y
x = 3
3 x (y = 0)
y = number: This is a horizontal line parallel to the x -axis.
The gradient is zero.
y
y = 3
x
• Parallel lines:
If AB // CD then m = m
AB CD
Example 1:
If y = 2x + 4 is parallel to y = ax – 2, find the value of a.
Solution:
If lines are parallel, the gradients are equal. ∴ a = 2
• Perpendicular lines:
If AB ⊥ CD then m × m = –1
AB CD
Example 2:
y = 2x + 4 is perpendicular to y = ax – 2.
Find the value of a.
Solution:
If lines are perpendicular (90°) to each other, the product
of their gradients will be equal to –1.
∴×mm=1
12
2m × =1
2
m 1
2 =
2
∴ If one gradient is given, the other gradient will have the
opposite sign and reciprocal (inverse) of the given
gradient.
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Brainline Mathematics Gr.12 Mod2-Unit AG 1 Revise Analytical Geometry P.3
Example 3:
Study the gradients in this table, if the lines are parallel and if
they are perpendicular to the each other.
Line Gradient if Gradient if
parallel to line perpendicular to line
y = 3x – 4 3 1
3
y= 2x + 1 2 +3
3 3 2
y = x + 2 1 –1
2y + x – 6 = 0
1 2
∴Write in
standard form: 2
2y = – x + 6
x
∴y = + 3
2
• Collinear points:
If A, B and C are collinear then m = m = m
AB BC AC
(gradients are equal and they share a common point).
Example:
Are K(2; 5), L(5; –1) and J(6; –3) collinear?
Solution:
For the points to be collinear, the gradients must be equal
y and a common point must be shared. Draw a rough sketch
to get an idea of what this looks like.
K(2; 5) If m = m , the common point is L.
KL LJ
yy yy
m= LJ m= kL
and
x LJ xx KL
x x
LJ
L(5; –1) KL
1 ( 3)
J(6; –3) = = 5 ( 1)
5 6
2 5
= 2 6
=
1
3
= 2 = 2
m = m and they share a common point L, so these points
KL LJ
are collinear. You could also have proved m = m , in which
KL KJ
case K would have been the common point.
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Brainline Mathematics Gr.12 Mod2-Unit AG 1 Revise Analytical Geometry P.4
To find the place where θ lies: • Inclination of line:
Put your pencil’s point on the x
-intercept, with the rest of the If θ is the angle between the positive x-axis and the line AB,
pencil lying on the positive x-axis. then tan θ = mAB y
That is the pl .
ace where θ begins y
Now keep the point on the
x-intercept and turn the pencil anti- θ x θ
clockwise, to the place where the x
line whose inclination you want to
find lies, then that turn of the pencil
is where θ lies.
If m > 0 (positive) then θ is If m < 0 (negative) then θ
is an obtuse angle.
an acute angle.
(θ = 180° – reference angle)
Let us work through a few examples covering Grade 10 and 11
Analytical Geometry problems.
In Analytical Geometry we usually Example 1:
draw rough sketches to get an idea
of what the question states. In the Consider the following points on a Cartesian plane: P(1; 2)
case of variables in a coordinate, Q(3; 1) , R(– 3; k) and S(2; –3). Determine value(s) of k if:
take an educated guess of where a) T(–1; 3) is the midpoint of PR.
this point will lie.
b) PQ // RS
c) PQ ⊥ RS
d) P, Q and R are collinear
52
e) RS =
Solution:
y
R(–3;k) If T is the midpoint of PR then
RT should lie to the left of point
T(–1;3) P(1;2) T. Indicate it on your graph.
Q(3;1)
x
Midpoint formula: a) P(1; 2); R (– 3; k) and T(– 1; 3)
x + + xyy
PRPR
; y + y
22 PR
y =
T 2
3 = 2 + k
2
6 = 2 + k
k = 4
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