Investigation.

1
Investigation.
Find the distance between two points A(1, 2) and
B(3, 6)
6
B(3,6)
Form a triangle and use Pythagoras to find the
distance between the points
(6 2 )
4
2
2
A(1,2)
x-length 3 1 2
3
1
y-length 6 2 4
(3 1 )
Length v(22 42 v20
2
Co-ordinate GeometryChapter 7
3
Note 1 Distance between two points
? B (x2 , y2)
Let A (x1 , y1) and B (x2 , y2) be two points.
D
A (x1 , y1) ?
y2 y1
x2 x1
We often need to find d, the distance between A
and B. This is found using PYTHAGORAS
4
Example
Find the distance between the points (3,2) and
(3,-6)
(x1, y1)
(x2, y2)
substitute into the formula
d 10 units
5
Applications
Prove that the vertices A(1, 5), B(2, 9) and C(6,
10) are those of an isosceles triangle. AB
v17 BC v17 AC v50 Because there is two
sides with the same length the triangle is
isosceles.
6
Theata Page 128 Exercise 16.2
7
Investigation
Consider the two numbers 6 and 10.
The number exactly halfway between them, their
MIDPOINT, is 8.
The midpoint can easily be worked out by counting
inwards from 6 and 10, but you can also find the
midpoint by averaging the two numbers.
This averaging is a really useful process when
the numbers are not as easy to work with as 6 and
10
8
This concept can now be used to find the midpoint
of two points on an x-y graph.
Example.
Find the midpoint of ( 3, 4) and (1, 2)
Step 1
Average the xs
(1,2)
Step 2
Average the ys
(1,1)
Step 3
M (1,1)
(3,4)
9
Note 2 Finding a Midpoint
Let A (x1 , y1) and B (x2 , y2) be two points.
? B (x2 , y2)
? M (? , ?)
? A (x1 , y1)
A general formula that finds the midpoint of any
two points is
10
Example

• Find the midpoint of the line segment joining
• E (10, -3) and F (6, 0)

(6 , 0) (x1 , y1)?
? (10,-3) (x2 , y2)
M (8, -1.5)
11
Prove that the points A(-1,-2), B(1, 1), C(8,-1)
and D(6,-4) are the vertices of a parallelogram.
(HINT find the lengths of all four sides and the
diagonals) AB 3.6 units BC 7.3 units DC
3.6 units AD 7.3 units AC 9.1 units BD
7.1 units Length of AB length of CD and length
of BC length of AD. The lengths of the
diagonals are different, so therefore the shape
is a parallelogram
12
Theata Page 127 Exercise 16.1
13
Starter
Find the midpoints of the line segments joining
(6, -2) and (1, 2) (3, -2) and (1, -6)
midpoint (3½, 0) midpoint (2, -4)
14
Note 3 Gradient (slope)

• measures the steepness of a line

15
Examples
Leans left, so m is negative!
4 cm
8
2 cm
8
3
2
5 km
3 km
16
C
y2
? B (x2 , y2 )
Let A (x1 , y1 ) and B (x2 , y2 ) be any two
points
? A (x1 , y1 )
y1
The GRADIENT of AB is given by
m
x2
x1
rise is y2 y1 run is x2 x1
17
Example
(x1, y1)
A(4,3)
Find the gradient of the line joining points A(4,
3) and B(1, 3)
M

C
B(1, 3)
(x2, y2)
18
Parallel and Perpendicular lines
Parallel lines have the same gradient Perpendicul
ar lines gradients are the negative reciprocals
of each other m1 x m2 -1 Example If line
AB has a gradient of 2/3, and line CD is
perpendicular to line AB, what is the gradient of
CD? Gradient of CD -3/2 Points are collinear
if they lie on the same line their gradients
are equal
19
Page 137 Exercise 7A
20
Note 4 Revision of Equations of Lines
The gradient/intercept form for the equation of a
line is y mx c
gradient
y-intercept
21
Example
Plot y -2/3x 2 using the gradient and
intercept method y-intercept (c) 2 Gradient (m)
-2 3
Fall is 2
Run is 3
Negative sign means it leans to LEFT

• Plot the y-intercept
• Plot the next point by making a triangle, whose
  fall is 2 and run is 3
• Plot another point in the same manner
• Connect the points with a straight line

22
Another option up (2) then left (-3)
?
One option down (-2) then right (3)
Now join the dots
?
y -2/3x 2
Dont forget arrows label
23
Exercises
Plot the following graphs y 2x 3 Y -?x
2 y x
24
The general form of the equation of a line
is ax by c 0 Equations can be
rearranged from gradient-intercept form to the
general form by performing operations on both
sides of the equation. (a is usually positive)
25
Example
Write the following equation in the general form
y -2/3x 2 3y -2x 6
2x 3y 6 0
Multiply equation by 3
Move everything to the LHS
26
Exercises
Write the following equations in the general
form y 4x 3 Y -?x 2 y ?x ?
4x y - 3 0
2x 5y - 10 0
x - 6y - 5 0
27
Note 5 Finding Equations of Lines
If you know

• the gradient, m

and

• any point, (x1 , y1)

then the equation of the line can be worked out
using the formula
y y1 m (x x1)
28
Find the equation of the line passing through (
3, 5) and with gradient 4
Example 1
Step 1
Write m, x1 and y1
m 4
x1 3
y1 5
Step 2
Put these values into the formula y y1 m(x
x1)
y 5 4(x 3)
Step 3
Remove brackets. Write in general form
y 5 4(x 3)
y 5 4x 12
0 4x - y 17
y 4x 17
29
Find the equation of the line passing through (
2, -13) and (3, 2)
Example 2
(x1, y1)
(x2, y2)
Step 1
Find m
m
3
Put these values into the formula y y1 m(x
x1)
Step 2
y -13 3(x 2)
Step 3
Remove brackets. Write in general form
Y 13 3(x 2)
Y 13 3x 6
0 3x - y - 7
y 3x - 7
30
Note 5a Finding Equations of Lines Quickly
For slope the form of the line is Ax By

For slope the form of the line is Ax By
Example Find the equation of the line which
passes through (2,-5) with a gradient of ? The
equation is 3x 8y 3(2) 8(-5) 3x 8y
46
31
Exercise 7B and C
32
Starter
A car club has an annual hill climb competition.
A cross section of part of the hill they race on
is drawn below. The beginning and end points of
the section, in metres from the start of the hill
are given as co-ordinate pairs.
33

• Find
• The gradient over this section of the hill climb
• What is the distance of this section of the hill
  climb in metres to the nearest metre?
• What are the co-ordinates of the half-way point
  of this section of the hill climb?

m 1/5
153m
(215, 85)
34
STARTERS
Graph the following lines 4x 12y 24
y - 2/3x 4
35
Note 6 Perpendicular Bisector
The perpendicular bisector of AB is the set of
all points which are the same distance from A and
B. The perpendicular bisector (or mediator) is a
line which is perpendicular to AB and passes
through the midpoint of AB.
36
Example Find the equation of the perpendicular
bisector between the points A(-2, 5) and B(4,
9) Find the midpoint of AB Find the gradient
of AB M The perpendicular gradient
of AB - 3/2 Equation of perpendicular
bisector
M (1, 7)
?
y 7 - 3/2 (x 1)
2y 14 - 3 (x - 1)
2y 14 - 3x 3
3x 2y 17 0
37
Page 146 Exercise 7D.1 and 7D.2
38
Starter

• Find the equation of the mediator of AB for the
  triangle A(-2,3), B(4,0) C(-2,-3)
• Find the midpoint of AB
• Find the gradient of AB
• Find the perpendicular gradient
• Solution
• Midpoint (1, 1.5)
• Gradient -0.5
• Perpendicular gradient 2
• Equation y 1.5 2( x 1 )
• 2x 2
• y 2x 0.5

A
midpoint
B
C
39
Starter

• 3 darts are thrown at a dart board. The first
  land in the bulls eye, the 2nd lands 2cm to the
  left and 1 cm above the bulls eye, the 3rd lands
  2cm to the right and 9cm above the bulls eye.
• Calculate the distance between the points of the
  2nd and 3rd darts
• The equation of the line joining these two darts
• Find the intersection of this line with its
  perpendicular bisector
• Find the equation on this perpendicular bisector

40
Other Geometrical Terms

• Equidistant find the midpoint
• Bisects cuts into two equal parts
• Perpendicular at right angles
• Vertex corner of angle
• Concurrent pass through the same point
• Collinear lies on the same line

41
Note 7 Triangles Altitude of Triangles

• The perpendicular distance from a vertex to the
  opposite side of a triangle is called the
  altitude (or height)
• Example In the triangle A(-4,4), B(2,2) and
  C(-2,-1), calculate the length of the altitude of
  the triangle ABC through vertex C.
• Find the intersection of CI and AB
• Find equation of AB
• Find equation of CI
• Solve a and b simultaneously

42

• Example In the triangle A(-4,4), B(2,2) and
  C(-2,-1), calculate the length of the altitude of
  the triangle ABC through vertex C.
• Find the intersection of CI and AB
• Find equation of AB
• Find equation of CI
• Solve a and b simultaneously

43
Note 7 Triangles Medians of Triangles
A line drawn from a vertex of a triangle to a
midpoint of the opposite side is called the median
44
Note 7 Triangles Medians of Triangles