1B_Ch8(1)
8.1 Rectangular Coordinates
1B_Ch8(2)
A Introduction to
Coordinate Systems
B Rectangular Coordinate
System
Index
8.2 Distances and Areas in the
1B_Ch8(3)
Rectangular Coordinate System
A Distance between Two
Points on a Horizontal or
Vertical Line
B Area of a Plane Figure
Index
8.3 Polar Coordinates
1B_Ch8(4)
A Introduction to
Polar Coordinates
B Comparison between
Rectangular and Polar
Coordinates
Index
8.1 Rectangular Coordinates
1B_Ch8(5)
 Example
A) Introduction to Coordinate Systems
‧
Refer to the following figure. A building is located in
the
area.
We use D3 to represent
its position.
This kind of method for representing positions is called
a coordinate system.
 Index 8.1
Index
8.1 Rectangular Coordinates
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The figure shows the seating plan of Class 1A. It is known that the
position of Ann is D2, indicate the position of Lily and James.
5
Lily
4
3
James
2
Ann
1
A
B
C
D
E
The position of Lily is B4 and the position of James is E3.
 Key Concept 8.1.1
Index
8.1 Rectangular Coordinates
1B_Ch8(7)
B) Rectangular Coordinate System
1. Ordered Pairs
‧
An ordered pair is a pair of numbers written within
brackets in a particular order.
E.g. (1, 2), (7, –5)
Index
8.1 Rectangular Coordinates
1B_Ch8(8)
B) Rectangular Coordinate System
2. In a rectangular coordinate plane, we can locate the
position of a point by its distances from the horizontal
x-axis and vertical y-axis. Its position can be written as
an ordered pair (a, b).
y
3. In the figure, the ordered pair (a, b)
denotes the coordinates of P where a is
a
b
called the x-coordinate, b is called the
y-coordinate.
P(a, b)
b
O
a
x
Index
8.1 Rectangular Coordinates
1B_Ch8(9)
 Example
B) Rectangular Coordinate System
y
b
O
P(a, b)
a
x
4. The intersection O(0, 0) of the x-axis and the y-axis is
called the origin, which is the reference point of all
points in the plane.
 Index 8.1
Index
8.1 Rectangular Coordinates
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What are the coordinates of the origin
O and the point C in the figure?
The coordinates of O are (0, 0).
The coordinates of C are (–4, –3).
Index
8.1 Rectangular Coordinates
1B_Ch8(11)
Write down the coordinates of the point P in each of the
following rectangular coordinate plane.
(a)
(b)
Fulfill Exercise Objective
(a) The coordinates of P are (3, 1).
(b) The coordinates of P are (–0.7, 7).
 Write down the coordinates of
given points in a rectangular
coordinate plane.
Index
8.1 Rectangular Coordinates
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(a) Mark the four points A(–6, 7), B(1, 0), C(–13, –4)
and D(2, 11) in the rectangular coordinate plane.
(b) Draw a line through A and B and another line
through C and D. What are the coordinates of the
point of intersection?
Index
8.1 Rectangular Coordinates
1B_Ch8(13)
 Back to Question
(a), (b)
Fulfill Exercise Objective
 Find the coordinates of
the point of intersection.
From the graph, the required coordinates are (–4, 5).
 Key Concept 8.1.2
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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A) Distance between Two Points on a Horizontal or Vertical
Line
(a) Any two points on the same
y
AB = x2 – x1
horizontal line have the same
A
y-coordinate. If A(x1, y) and B(x2, y)
are these two points and x2 > x1,
O
B
x
then
AB = x2 – x1.
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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 Example
A) Distance between Two Points on a Horizontal or Vertical
Line
(b) Any two points on the same vertical
y
P
line have the same x-coordinate.
PQ = y2 – y1
If P(x, y1) and Q(x, y2) are these two
points and y2 > y1,
Q
O
x
then
PQ = y2 – y1.
 Index 8.2
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(16)
Find the lengths of the line segments shown in the diagram.
y
M(–11, 6)
N(2, 6)
T(–7, 3)
O
x
S(–7, –3)
MN = [2 – (–11)] units
= 13 units
TS = [3 – (–3)] units
= 6 units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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A(–15, 30), B(–15, –20), C(55, –20) and D(55, 30) are
four points in a rectangular coordinate plane. Given that
ABCD is a rectangle, what is its perimeter?
AB = [30 – (–20)] units
= 50 units
BC = [55 – (–15)] units
= 70 units
∴ Perimeter of ABCD = (AB + BC) × 2
= (50 + 70) × 2 units
Fulfill Exercise Objective
 Find the lengths of the sides
or the perimeters of figures.
= 240 units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
In the figure, if AB = 7 units,
1B_Ch8(18)
y
B(b, 3)
A(4, 3)
find the value of b.
O
x
Since A and B have the same y-coordinate (i.e. 3),
AB is horizontal.
Fulfill Exercise Objective
From the figure, 4 > b
∴4–b =7
b = –3
 Given the distance between two points
on the same horizontal or vertical line,
find the coordinates or the unknown in
the coordinates of a certain point.
 Key Concept 8.2.1
Index
8.2 Distances and Areas in the Rectangular Coordinate System
B)
1B_Ch8(19)
Area of a Plane Figure
1. We can find the areas of geometric figures in a
rectangular coordinate plane by finding the lengths
of some suitable vertical or horizontal line
segments.
 Example
2. Sometimes, indirect methods such as splitting or
combining figures may be needed.
 Example
 Index 8.2
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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The figure shows a triangle with vertices at A(–5, 6), B(–5, –2) and
C(8, –2). Calculate the area of △ABC.
y
A(–5, 6)
6
5
4
3
2
1
0
–7 –6 –5 –4 –3 –2 –1–1
–2
B(–5, –2)–3
x
1 2
3 4 5
6 7 8
9 10
C(8, –2)
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(21)
 Back to Question
Base = BC
Height = AB
= [8 – (–5)] units
= [6 – (–2)] units
= 13 units
= 8 units
Area of △ABC =
1
× BC × AB
2
1
= × 13 × 8 sq. units
2
= 52 sq. units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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The figure shows a triangle
with vertices at H(–1, –2),
K(–1, 2) and G(4, 3).
(a) Find the length of HK.
 Soln
(b) Find the height of △HKG with respect to the base HK.
(c) Calculate the area of △HKG.
 Soln
 Soln
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(23)
 Back to Question
(a) HK = [2 – (–2)] units
= 4 units
(b) Through G, construct a perpendicular
to HK to meet HK produced at N.
The coordinates of N are (–1, 3).
When HK is the base,
height = GN
= [4 – (–1)] units
= 5 units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(24)
 Back to Question
1
× HK × GN
2
1
= × 4 × 5 sq. units
2
(c) Area of △ HKG =
= 10 sq. units
Fulfill Exercise Objective
 Find areas of simple figures.
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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In the figure, A(1, 2), B(–2, –2),
C(4, –2) and D(3, 2) are the four
vertices of a trapezium. Find the
area of ABCD.
AD = (3 – 1) units
= 2 units
BC = [4 – (–2)] units
= 6 units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(26)
 Back to Question
Through A, construct a perpendicular AE to BC.
The coordinates of E are (1 , –2).
AE = [2 – (–2)] units
= 4 units
1
× (AD + BC) × AE
2
1
= × (2 + 6) × 4 sq. units
2
∴ Area of ABCE =
= 16 sq. units
Fulfill Exercise Objective
 Find areas of simple figures.
 Key Concept
Index
8.2.2
8.2 Distances and Areas in the Rectangular Coordinate System
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In the figure, the vertices of the quadrilateral are K(–2, 5),
L(–5, –3), M(–2, –4) and N(4, –3). Find the area of the
quadrilateral.
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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 Back to Question
Join KM so that the figure is split into
△KLM and △KMN. Then draw line
segments LP and NP as shown.
From the figure,
the coordinates of P are (–2, –3).
1
Area of △KLM = × KM × LP
2
1
= × 9 × 3 sq. units
2
= 13.5 sq. units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(29)
 Back to Question
1
Area of △KMN = × KM × NP
2
1
= × 9 × 6 sq. units
2
= 27 sq. units
∴ Area of KLMN
= area of △KLM + area of △KMN
= (13.5 + 27) sq. units
= 40.5 sq.units
Fulfill Exercise Objective
 Find areas of composite figures by splitting figures.
Index
8.2 Distances and Areas in the Rectangular Coordinate System
Find the area of pentagon
y
R(–5, 6)
RMSTU in the figure.
P
6
5
4
3
2
1
0
–7 –6 –5 –4 –3 –2 –1–1
–2
S(–5, –2)–3
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U(8, 6)
M(0, 2)
x
1 2
3 4 5
6 7 8
9 10
T(8, –2)
Join RS so that the figure becomes a rectangle RSTU.
Then draw line segment MP as shown.
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(31)
 Back to Question
1
Area of △RMS = × RS × MP
2
1
= × 8 × 5 sq. units
2
= 20 sq. units
∴ Area of RMSTU
= area of RSTU – area of △RMS
= [(13 × 8) – 20] sq. units
= (104 – 20) sq. units
= 84 sq. units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
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Find the area of quadrilateral
OPQR in the figure.
Draw two perpendiculars PA and QB
to the x-axis.
Then APQB is a trapezium, where the
coordinates of A are (–3, 0) and the
coordinates of B are (8, 0).
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(33)
 Back to Question
1
Area of trapezium APQB = × (AP + BQ) × AB
2
1
= × (4 + 6) × 11 sq. units
2
= 55 sq. units
1
Area of △OAP = × OA × AP
2
1
= × 3 × 4 sq. units
2
= 6 sq. units
Index
8.2 Distances and Areas in the Rectangular Coordinate System
1B_Ch8(34)
 Back to Question
1
Area of △RBQ = × RB × BQ
2
1
= × 6 × 6 sq. units
2
= 18 sq. units
∴ Area of OPQR
= area of trapezium APQB – area of △OAP –
area of △RBQ
= (55 – 6 – 18) sq. units
= 31 sq. units
Fulfill Exercise Objective
 Find areas of figures by subtraction.
 Key Concept 8.2.2
Index
8.3 Polar Coordinates
A)
1B_Ch8(35)
Introduction to Polar Coordinates
1. In the polar coordinate plane in the figure, the point P is
r units from the pole O. The angle which is measured
anticlockwise from the polar axis OX to OP is θ. We can
locate the position of P by r and θ, expressed as the
ordered pair (r, θ).
Index
8.3 Polar Coordinates
1B_Ch8(36)
 Example
A) Introduction to Polar Coordinates
2. The ordered pair (r, θ) denotes the polar coordinates of P,
where r is the radius vector and θ is the polar angle.
 Index 8.3
Index
8.3 Polar Coordinates
1B_Ch8(37)
Write down the polar coordinates of the points M and N in the
given polar coordinate plane.
(a)
(b)
N
M
8
3
85
O
X
O
105
X
(a) The polar coordinates of M are (8, 85).
(b) The polar coordinates of N are (3, 105).
Index
8.3 Polar Coordinates
1B_Ch8(38)
Write down the polar
coordinates of the points A,
B and C in the given polar
coordinate plane.
The polar coordinates of A are (4, 40).
The polar coordinates of B are (3, 140).
The polar coordinates of C are (5, 240).
Fulfill Exercise Objective
 Write down the polar
coordinates of points.
 Key Concept 8.3.1
Index
8.3 Polar Coordinates
1B_Ch8(39)
 Example
B) Comparison between Rectangular and Polar Coordinates
3. It is easier to find the distance between any point and O
in a polar coordinate plane than in a rectangular
coordinate plane. However, it is often difficult to find the
distance between two points on a vertical or horizontal
line in a polar coordinate plane.
 Index 8.3
Index
8.3 Polar Coordinates
1B_Ch8(40)
Which figures you will use if measuring the length of OA and AB?
y
4
3
A
A
2
1
–5
–4 –3 –2 –1 O
–1
1
2
3
4
5
x
–2
B
–3
B
–4
–5
Fig. I
Fig. II
We use Fig. I to measure the length of OA and use
Fig. II to measure the length of AB.
 Key Concept 8.3.2
Index