# Coordinate Geometry 2

## Transcription

Coordinate Geometry 2
```Coordinate Geometry
Student
Book - Series J-1
(y1, y2)
(x1, x2)
Mathletics
Instant
Workbooks
Coordinate geometry
Student Book - Series J
Contents
Topics
Date completed
Topic 1 - Plotting points
__/__/__
Topic 2 - The distance formula
__/__/__
Topic 3 - The midpoint of an interval
__/__/__
Topic 4 - The gradient of a line
__/__/__
Topic 5 - Different forms of linear equations
__/__/__
Topic 6 - Parallel lines
__/__/__
Practice Tests
Topic 1 - Topic test A
__/__/__
Topic 2 - Topic test B
__/__/__
Author of The Topics and Topic Tests: AS Kalra
Coordinate geometry
Mathletics Instant Workbooks – Series J
CHAPTER 10
Coordinate
geometry
Coordinate Geometry
Topic 1: Plotting points
page 45
UNIT 1: Plotting points
Plot each pair of points on the number plane and find the distance between them.
QUESTION 1
a
EXCEL INTERMEDIATE MATHS YEARS 9–10
A(1, 2) and B(4, 2), AB = __________________________
b
C(1, 4) and D(3, 4), CD = __________________________
c
E(3, 1) and F(3, 5), EF = __________________________
d
G(2,1) and H(2, 2), GH = __________________________
e
I(4, 0) and J(4, 5), IJ = __________________________
f
K(1, 3) and L(5,3), KL = __________________________
g
M(1,5) and N(5, 5), MN = __________________________
h
Q(0, 1) and P(4, 1), QP = __________________________
6
y
5
4
3
2
1
0
1
2
3
4
5
x
QUESTION 2 Use Pythagoras’ theorem to find the distance AB in each diagram. Leave your answers
in surd (square root) form where necessary.
a
b
y
• A (4, 4)
c
y
y
• A (1, 3)
•
B (–2, 1)
• A (5, 3)
x
0
•
x
0
B (–3, 0)
x
0
•
B (–3, –2)
_______________________
________________________
_______________________
_______________________
________________________
_______________________
QUESTION 3 Use Pythagoras’ theorem to find the length of each interval. Leave your answers in surd form
where necessary.
a
b
y
y
M (3, 4)
P (–3, 4)
• A (1, 4)
0
c
y
•
x
•
x
0
L (–3, 0)
•
0
x
• Q (5, –2)
• B (5. –3)
_______________________
________________________
_______________________
_______________________
________________________
_______________________
91
Chapter 10: Coordinate Geometry
Coordinate geometry
Mathletics Instant Workbooks – Series J
1
Coordinate Geometry
Coordinate geometry
UNIT 2: The distance formula
EXCEL INTERMEDIATE MATHS YEARS 9–10
pages 50–51
Topic 2: The distance formula
QUESTION 1 Use the distance formula d = (x2 – x1 )2 + (y2 – y1 )2 to find the distance between the following
a
A(2, 5), B(7, 13) _________________________
b
_______________________________________
c
A(0, 1), B(3, –4) _________________________
_______________________________________
d
_______________________________________
e
A(4, 5), B(7, 9) __________________________
A(–1, –4), B(3, –8) _______________________
A(3, 2), B(6, 6) __________________________
_______________________________________
f
_______________________________________
A(5, –2), B(7, –5) ________________________
_______________________________________
QUESTION 2 Calculate the distance between the following pairs of points.
a
P(4, 3), Q(3, 2) __________________________
b
_______________________________________
c
P(–1, –3), Q(2, –5) _______________________
_______________________________________
d
_______________________________________
e
P(1, 3), Q(3, 5) __________________________
P(2, 5), Q(8, 12) _________________________
P(–3, 2), Q(1, –6) ________________________
_______________________________________
f
_______________________________________
P(4, –5), Q(6, –9) ________________________
_______________________________________
QUESTION 3 Find the perimeter of a triangle whose vertices are A(6, 2), B(5, 2) and C(–4, –5).
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
QUESTION 4 Find the distance between the points A(2, 5) and B(5, 10) and then square it.
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
92
EXCEL ESSENTIAL SKILLS: YEAR 9 MATHEMATICS REVISION AND EXAM WORKBOOK 1
Coordinate geometry
Mathletics Instant Workbooks – Series J
2
Coordinate Geometry
CoordinateUNIT
geometry
3: The midpoint of an interval
EXCEL INTERMEDIATE MATHS YEARS 9–10
pages 50–51
Topic 3: The midpoint of an interval
QUESTION 1 Use the midpoint formula x = x1 + x2, y = y1 + y2 to find the midpoint of the interval joining the
following points.
a
2
A(0, 6), B(2, 4) _________________________
2
b
_______________________________________
c
A(–3, 2), B(–5, 0) ________________________
_______________________________________
d
_______________________________________
e
A(7, 0), B(5, 0) __________________________
A(4, 8), B(6, 10) _________________________
A(3, 8), B(7, 2) __________________________
_______________________________________
f
_______________________________________
A(2, 10), B(4, 4) _________________________
_______________________________________
QUESTION 2 Find the midpoint of the interval joining the following points.
a
P(–4, –11), Q(7, 4) _______________________
b
_______________________________________
c
P(2, 10), Q(8, 8) _________________________
_______________________________________
d
_______________________________________
e
P(–3, –6), Q(1, 4) ________________________
P(10, 4), Q(8, 6) _________________________
P(4, 5), Q(6, 9) __________________________
_______________________________________
f
_______________________________________
P(–8, 2), Q(4, –6) ________________________
_______________________________________
QUESTION 3 The vertices of a triangle ABC are A(–2, 9), B(10, 11) and C(–7, 1). Find the midpoint of each
side.
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
QUESTION 4 Prove that the midpoint of (7, –3) and (–7, 3) is the origin.
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
93
Chapter 10: Coordinate Geometry
Coordinate geometry
Mathletics Instant Workbooks – Series J
3
Coordinate Geometry
Coordinate geometry
UNIT 4: The gradient of a line
EXCEL INTERMEDIATE MATHS YEARS 9–10
pages 46, 50
Topic 4: The gradient of a line
QUESTION 1 Use the gradient formula m = y2 – y1 to find the gradient of the straight line passing through the
x2 – x1
following.
a
(2, 6) and (1, –8) _________________________
b
_______________________________________
c
(–1, –2) and (3, –8) _______________________
_______________________________________
d
_______________________________________
e
(8, 7) and (2, 8) _________________________
(–3, –4) and (1, 2) ________________________
(4, 6) and (1, 5) __________________________
_______________________________________
f
_______________________________________
(8, 0) and (10, –1) _______________________
_______________________________________
QUESTION 2 Find the gradient of the line between
a
(5, 6) and (3, 2) _________________________
b
_______________________________________
c
(–4, 1) and (2, –5) ________________________
_______________________________________
d
_______________________________________
e
(–2, –1) and (–5, –3) ______________________
(–1, 1) and (2, 5) ________________________
(5, –2) and (2, 12) ________________________
_______________________________________
f
_______________________________________
(–2, 4) and (6, 3) ________________________
_______________________________________
QUESTION 3 Find correct to two decimal places, where necessary, the gradient of a line that is inclined to the
positive direction of the x-axis at an angle of
a
d
30° ____________________
b
45° ____________________
c
60° ___________________
________________________
________________________
_______________________
________________________
________________________
_______________________
120° ___________________
e
135° ___________________
f
150° __________________
________________________
________________________
_______________________
________________________
________________________
_______________________
QUESTION 4 Show that (–3, 4), (0, 2) and (6, –2) are collinear.
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
94
EXCEL ESSENTIAL SKILLS: YEAR 9 MATHEMATICS REVISION AND EXAM WORKBOOK 1
Coordinate geometry
Mathletics Instant Workbooks – Series J
4
Coordinate Geometry
Coordinate
geometry
UNIT 5: Different forms of linear equations
EXCEL INTERMEDIATE MATHS YEARS 9–10
page 49
Topic 5: Different forms of linear equations
QUESTION 1 Write each of the following equations in the general form.
a
3x + 5y = –9 _____________
b
________________________
d
6y + 8 = 5x ______________
y = 32 x – 5 _______________
c
________________________
e
________________________
g
x – 2y = 6 _______________
4x – y = 9 _______________
_______________________
f
________________________
h
________________________
9x – 6 = 7y ______________
4x – 5 = 3y _____________
y = 3x – 7 ______________
_______________________
i
________________________
y = – x5 + 2 ______________
_______________________
QUESTION 2 Write each of the following equations in the gradient–intercept form.
a
4y = 8x – 12 ____________
b
_______________________
d
y + 6x = 0 ______________
5x – 3y = 12 ____________
c
________________________
e
_______________________
g
8 + y = 5x _______________
8x – 3y = 15 _____________
_______________________
f
________________________
h
_______________________
5y = 9x – 12 _____________
7y – 4x = 11 ____________
2x + y = 1 ______________
_______________________
i
________________________
y + 3x = 15 _____________
_______________________
QUESTION 3 Write down the gradient (m) and the y-intercept (b) for each of the following.
a
y = 3x – 1 ______________
b
_______________________
d
y=
1
x
4
– 3 ______________
y = 9x – 7 _______________
c
________________________
e
_______________________
y=
3
x
5
– 6 _______________
y = 25 x – 6 _____________
_______________________
f
________________________
y = x __________________
_______________________
QUESTION 4 Write the equation of the line in the gradient–intercept form for each of the following when the
gradient (m) and the y-intercept (b) are given.
a
m = 3, b = 2 ____________
b
_______________________
d
m = 12 , b = 7 _____________
_______________________
m = 6, b = – 4 ____________
c
________________________
e
m = – 23 , b = – 4 ____________ f
________________________
m = –1, b = –3 __________
_______________________
m = –6, b = 4 ___________
_______________________
QUESTION 5 State whether the point given after each linear equation lies on that line.
a
x – 2y = 4 (0, –2) _______________________
b
_______________________________________
c
2x – 3y = 6 (3, 0) _______________________
_______________________________________
d
_______________________________________
e
3y – 4x = 12 (0, 4) ______________________
y = 5x – 6 (–1, 4) _______________________
y = 12 x – 4 (2, –3) _______________________
_______________________________________
f
_______________________________________
y = 34 x – 2 (8, 1) ________________________
_______________________________________
95
Chapter 10: Coordinate Geometry
Coordinate geometry
Mathletics Instant Workbooks – Series J
5
Coordinate Geometry
Coordinate geometry
UNIT 6: Parallel lines
EXCEL INTERMEDIATE MATHS YEARS 9–10
pages 53–54
Topic 6: Parallel lines
QUESTION 1 State whether or not each pair of lines is parallel.
a
x + 2y + 7 = 0 and x + 2y – 3 = 0 _________________________________________________________
b
3x – 3y + 5 = 0 and 3x – 2y – 9 = 0 _______________________________________________________
c
4x + y – 6 = 0 and 4x + y + 3 = 0 _________________________________________________________
d
x + 3y + 1 = 0 and x – 3y – 2 = 0 _________________________________________________________
e
4y = 3x – 5 and y = 3x + 7 ______________________________________________________________
f
y = 2x + 11 and y = 2x – 5 ______________________________________________________________
QUESTION 2 Change each equation to gradient–intercept form and then decide whether or not each pair of
lines is parallel.
a
2x – 3y = 7 and 3y = 2x + 5 ______________________________________________________________
b
–4x + 7 = 5y and 5y + 4x – 9 = 0 _________________________________________________________
c
2x – 8y + 5 = 0 and 8y = 2x + 8 __________________________________________________________
d
x – y + 8 = 0 and x + y + 2 = 0 ___________________________________________________________
e
6x – 2y = 9 and 3x = y – 3 _______________________________________________________________
f
2x – 5y = 7 and 5y = 2x – 8 ______________________________________________________________
QUESTION 3 What is the gradient of any line that is parallel to the following lines?
a
y = – 25 x + 8 _____________________________
b
3x – 4y + 6 = 0 __________________________
_______________________________________
_______________________________________
QUESTION 4 Find the gradient of a straight line parallel to the line joining (2, –3) and (7, 2).
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
QUESTION 5 Show that the line joining (4, –7) and (–2, 5) is parallel to the line 2x + y – 10 = 0.
_________________________________________________________________________________________
_________________________________________________________________________________________
QUESTION 6 Only two of the following lines are parallel. Find them.
y = 2x + 3, 2x + y = 3, 2x – y = 3, 3x + y = 2, x + 2y = 3.
_________________________________________________________________________________________
_________________________________________________________________________________________
96
EXCEL ESSENTIAL SKILLS: YEAR 9 MATHEMATICS REVISION AND EXAM WORKBOOK 1
Coordinate geometry
Mathletics Instant Workbooks – Series J
6
Maths Revision Yr 9 Part 1.qxd:2717_0108R1_pp01_98.qxd
21/2/08
6:40 PM
Page 97
Coordinate geometry
TOPICTest
TEST
Topic
PARTAA
PART
Coordinate
Geometry
Time allowed: 15 minutes
Total marks = 15
Time allowed: 15 minutes
1
The point (6, 2) lies on the line
A 2x – 3y = 6
B 2x + 3y = 6
2
1
D
–10
1
The midpoint of the interval joining the points (5, 9) and (–7, 1) is
A (1, –5)
B (–1, –5)
C (1, 5)
D
(–1, 5)
1
Write 2y – 3x = 8 in the gradient–intercept form.
A y = 32 x – 4
B y = 32 x + 4
y = – 32 x + 4
D
y = – 32 x – 4
1
What is the gradient of a line that is parallel to the line 2y = 6x – 7?
A 6
B –6
C 3
D
–3
1
The line y = 3x passes through the point
A (0, –1)
B (0, 0)
D
(0, 2)
1
D
5
1
D
(0, 7)
1
D
13
1
9
8
Find the equation of the line in the gradient–intercept form when the gradient (m) is 12 and the
y-intercept (b) is –5.
A y = – 12 x + 5
B y = 12 x + 5
C y = 12 x – 5
D y = – 12 x – 5
7
3x + 2y = 6
1
6
D
Find the length of the interval AB joining the points A(2, 5) and (8, 13).
A 18
B –
18
C 10
5
3x – 2y = 6
1
4
C
Marks
What is the gradient of the line that passes through the points (1, 3) and (2, –5)?
A –1
B 1
C –8
D 8
3
Total marks = 15
C
C
(0, 1)
Find the distance between the origin and the point (3, 4).
A 7
B 5
C 7
10 Find the coordinate of the midpoint of (–3, 7) and (3, –7).
A
(3, 7)
B
C
(–3, –7)
(0, 0)
11 A line passes through the points A(–2, 4) and B(3, 9). Find the gradient.
A
–1
B
C
1
–13
12 Find the gradient of a line that is inclined to the positive direction of the x-axis at an angle of 45°.
A
–1
B
C
1
– 12
D
1
2
13 Find the angle of inclination to the positive direction of the x-axis of a line with gradient 3.
A
30°
B
45°
14 Write y = 2x – 3 in general form.
A
y – 2x = 3
B
2x – y = 3
C
60°
D
75°
1
C
2x – y – 3 = 0
D
y – 2x + 3 = 0
1
D
5
1
15 Find the gradient of a straight line parallel to the line 3x – y + 5 = 0.
A
1
3
B
3
C
1
5
Total marks
for PART
PART A
Totalachieved
marks for
A
Chapter
Chapter 10:
10: Coordinate
Coordinate Geometry
Geometry
1
Coordinate geometry
Mathletics Instant Workbooks – Series J
15
15
977
Maths
Maths Revision
Revision Yr
Yr 99 Part
Part 1.qxd:2717_0108R1_pp01_98.qxd
1.qxd:2717_0108R1_pp01_98.qxd
21/2/08
21/2/08
6:40
6:40 PM
PM
Page
Page 98
98
Coordinate geometry
TOPIC TEST
Topic Test
PART B
PART B
Coordinate Geometry
Time allowed: 15 minutes
Total marks = 15
Time
Time allowed:
allowed: 15
15 minutes
minutes
Total
Total marks
marks == 15
15
Marks
Question
Question 1
1
aa
Use
Use Pythagoras’
Pythagoras’ theorem
theorem to
to find
find the
the
length
length of
of aa diagonal
diagonal of
of aa 44 cm
cm by
by 44 cm
cm
square
square (correct
(correct to
to one
one decimal
decimal place).
place).
___________________________
___________________________
11
bb
Find
Find the
the distance
distance between
between the
the points
points A(–2,
A(–2, 4)
4) and
and B(2,
B(2, 7)
7)
and
and then
then square
square it.
it.
___________________________
___________________________
11
cc
Find
Find the
the exact
exact distance
distance between
between the
the origin
origin and
and the
the point
point
(4,
(4, –2).
–2).
___________________________
___________________________
11
dd
Prove
Prove that
that the
the midpoint
midpoint of
of (3,
(3, –4)
–4) and
and (–3,
(–3, 4)
4) is
is the
the origin.
origin.
___________________________
___________________________
11
ee
The
The coordinates
coordinates of
of the
the midpoint
midpoint of
of AB
AB are
are (2,
(2, 3).
3). If
If A
A is
is the
the
point
point (–3,
(–3, –5),
–5), what
what are
are the
the coordinates
coordinates of
of B?
B?
___________________________
___________________________
11
Question
Question 2
2 The
The equation
equation of
of aa straight
straight line
line is
is 3x
3x == yy –– 6.
6.
aa
Write
Write the
the equation
equation in
in the
the general
general form.
form.
___________________________
___________________________
11
bb
Write
Write the
the equation
equation in
in the
form.
___________________________
___________________________
11
cc
What
What is
is the
of this
this line?
line?
___________________________
___________________________
11
dd
What
What is
is the
the y-intercept
y-intercept of
of this
this line?
line?
___________________________
___________________________
11
ee
Is
Is this
this line
line parallel
parallel to
to the
the line
line 2y
2y == 6x
6x ++ 9?
9?
___________________________
___________________________
11
Question
Question 3
3
aa
Show
Show that
that (0,
(0, –5),
–5), (4,
(4, 5)
5) and
and (–2,
(–2, –10)
–10) are
are collinear.
collinear.
___________________________
___________________________
11
bb
Find the
the value
value of
of x.
x.
The
of (3,
(3, –1)
–1) and
and (x,
(x, –2)
–2) is
is ––1616.. Find
___________________________
___________________________
11
cc
Find
Find the
of aa straight
straight line
line parallel
parallel to
to the
the line
line
4x
4x –– 3y
3y ++ 99 == 0.
0.
___________________________
___________________________
11
dd
What
What is
is the
the value
value of
of kk ifif the
the lines
lines yy == 5x
5x –– 33 and
and 2y
2y == kx
kx ++ 77
are
are parallel?
parallel?
___________________________
___________________________
11
ee
Write
Write in
in general
general form
form the
the equation
equation of
of the
the line
line that
that has
has aa
y-intercept
y-intercept of
of –3
–3 and
and is
is parallel
parallel to
to the
the line
line yy == 7x
7x ++ 2.
2.
___________________________
___________________________
11
Total
Total marks
marks for
for PART
PART B
B
Total marks achieved for PART B
98
98
15
15
15
EXCEL
EXCEL ESSENTIAL
ESSENTIAL SKILLS:
SKILLS: YEAR
YEAR 99 MATHEMATICS
MATHEMATICS REVISION
REVISION AND
AND EXAM
EXAM WORKBOOK
WORKBOOK 11
Coordinate geometry
Mathletics Instant Workbooks – Series J
8
```