1b_ch8(1). 8.1rectangular coordinates a introduction to coordinate systems b rectangular coordinate...

1B_Ch8(1)

8.1 Rectangular Coordinates

A Introduction to

Coordinate Systems

B Rectangular Coordinate

System

Index

1B_Ch8(2)

8.2 Distances and Areas in the Rectangular Coordinate System

Index

1B_Ch8(3)

A Distance between Two Points on a Horizontal or Vertical Line

B Area of a Plane Figure

8.3 Polar Coordinates

A Introduction to

Polar Coordinates

B Comparison between

Rectangular and Polar

Coordinates

Index

1B_Ch8(4)

Introduction to Coordinate Systems

Index

8.1 Rectangular Coordinates 1B_Ch8(5)

A)

We use D3 to represent

its position.

This kind of method for representing positions is called a coordinate system.

Example

Index 8.1

‧ Refer to the following figure. A building is located in

the area.

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.

Key Concept 8.1.1 Index


Ann

5

4

3

2

1A B C D E

Lily

James

The position of Lily is B4 and the position of James is E3.

Rectangular Coordinate System

1. Ordered Pairs

Index


B)

‧ An ordered pair is a pair of numbers written

within brackets in a particular order.

E.g. (1, 2), (7, –5)


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).

Index


B)

O

b

a

P(a, b)

y

x

a

b

3. In the figure, the ordered pair (a, b)

denotes the coordinates of P where a is

called the x-coordinate, b is called the

y-coordinate.


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


B)

O

b

a

P(a, b)

y

x

Example

Index 8.1

What are the coordinates of the origin

O and the point C in the figure?

Index


The coordinates of O are (0, 0).

The coordinates of C are (–4, –3).

Write down the coordinates of the point P in each of the

following rectangular coordinate plane.

Index


(a) (b)

(a) The coordinates of P are (3, 1).

(b) The coordinates of P are (–0.7, 7).

Fulfill Exercise Objective

Write down the coordinates of given points in a rectangular coordinate plane.

(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


Index


(a), (b)

From the graph, the required coordinates are (–4, 5).


Find the coordinates of the point of intersection.

Key Concept 8.1.2

Back to Question

Distance between Two Points on a Horizontal or Vertical

Line

Index

8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(14)

A)

(a) Any two points on the same

horizontal line have the same

y-coordinate. If A(x1, y) and B(x2, y)

are these two points and x2 > x1,

then AB = x2 – x1.

O

y

xA B

AB = x2 – x1

Distance between Two Points on a Horizontal or Vertical

Line

Index


A)

(b) Any two points on the same vertical

line have the same x-coordinate.

If P(x, y1) and Q(x, y2) are these two

points and y2 > y1,

thenPQ = y2 – y1.

O

y

xQ

P

PQ = y2 – y1

Example

Index 8.2

Find the lengths of the line segments shown in the diagram.

Index


Ox

yM(–11, 6) N(2, 6)

T(–7, 3)

S(–7, –3)

MN = [2 – (–11)] units

= 13 units

TS = [3 – (–3)] units

= 6 units

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?

Index


∴ Perimeter of ABCD

AB = [30 – (–20)] units

= 50 units

BC = [55 – (–15)] units

= 70 units

= (AB + BC) × 2

= (50 + 70) × 2 units

= 240 unitsFulfill Exercise Objective

Find the lengths of the sides or the perimeters of figures.

In the figure, if AB = 7 units,

find the value of b.

Index


O

A(4, 3)B(b, 3)y

x

Since A and B have the same y-coordinate (i.e. 3),

AB is horizontal.

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

Area of a Plane Figure

Index


B)

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

The figure shows a triangle with vertices at A(–5, 6), B(–5, –2) and

C(8, –2). Calculate the area of △ABC.

Index


–7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10x

y

654321

–1–2–3

0

A(–5, 6)

B(–5, –2) C(8, –2)

Index


Area of △ABC = × BC × AB21

= 52 sq. units

= × 13 × 8 sq. units21

Base = BC

= [8 – (–5)] units

= 13 units

Height = AB

= [6 – (–2)] units

= 8 units

Back to Question

The figure shows a

triangle with vertices at

H(–1, –2), K(–1, 2) and

G(4, 3).

Index


(a) Find the length of HK.

(b) Find the height of △HKG with respect to the base HK.

(c) Calculate the area of △HKG.

Soln

Soln Soln

Index


= [2 – (–2)] units

= 4 units

(a) HK

(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

Back to Question

Index


= 10 sq. units

(c) Area of △ HKG = × HK × GN21

= × 4 × 5 sq. units21


Find areas of simple figures.

Back to Question

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.

Index


= (3 – 1) units

= 2 units

AD = [4 – (–2)] units

= 6 units

BC

Index


Through A, construct a perpendicular AE to BC.

The coordinates of E are (1 , –2).

= [2 – (–2)] units

= 4 units

AE

= 16 sq. units

∴ Area of ABCE = × (AD + BC) × AE21

= × (2 + 6) × 4 sq. units21

Back to Question

Key Concept

8.2.2


Find areas of simple figures.

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


Index


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).

Area of △KLM

= 13.5 sq. units

= × KM × LP21

= × 9 × 3 sq. units21

Back to Question

Index


Area of △KMN

= 27 sq. units

= × KM × NP21

= × 9 × 6 sq. units21

∴ Area of KLMN = area of △KLM + area of △KMN

= (13.5 + 27) sq. units

= 40.5 sq.units


Find areas of composite figures by splitting figures.

Back to Question

Find the area of pentagon

RMSTU in the figure.

Index


Join RS so that the figure becomes a rectangle RSTU.

Then draw line segment MP as shown.

–7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10x

y

654321

–1–2–3

0

R(–5, 6)

S(–5, –2)

U(8, 6)

T(8, –2)

M(0, 2)P

Index


Area of △RMS

= 20 sq. units

= × RS × MP21

= × 8 × 5 sq. units21

∴ Area of RMSTU = area of RSTU – area of △RMS

= [(13 × 8) – 20] sq. units

= (104 – 20) sq. units

= 84 sq. units

Back to Question

Find the area of quadrilateral

OPQR in the figure.

Index


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


Area of trapezium APQB

= 55 sq. units

= × (AP + BQ) × AB21

= × (4 + 6) × 11 sq. units21

Area of △OAP

= 6 sq. units

= × OA × AP21

= × 3 × 4 sq. units21

Back to Question

Index


Area of △RBQ

= 18 sq. units

= × RB × BQ21

= × 6 × 6 sq. units21

∴ Area of OPQR = area of trapezium APQB – area of △OAP –

area of △RBQ

= (55 – 6 – 18) sq. units

= 31 sq. units


Find areas of figures by

subtraction.

Key Concept 8.2.2

Back to Question

Introduction to Polar Coordinates

Index

8.3 Polar Coordinates 1B_Ch8(35)

A)

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).

Introduction to Polar Coordinates

Index


A)

2. The ordered pair (r, θ) denotes the polar coordinates of P,

where r is the radius vector and θ is the polar angle.

Example

Index 8.3

Write down the polar coordinates of the points M and N in the

given polar coordinate plane.

Index

(a) The polar coordinates of M are (8, 85).


(a) (b)

O X85

M

8

O X105

N

3

(b) The polar coordinates of N are (3, 105).

Write down the polar

coordinates of the points A,

B and C in the given polar

coordinate plane.

Index

1B_Ch8(38)

The polar coordinates of A are (4, 40).

The polar coordinates of B are (3, 140).

The polar coordinates of C are (5, 240). Key Concept 8.3.1

8.3 Polar Coordinates


Write down the polar coordinates of points.

Comparison between Rectangular and Polar Coordinates

Index


B)

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.

Example

Index 8.3

Index


Fig. I

A

B

x

y

–5 –4 –3 –2 –1 O 1 2 3 4 5

4

3

2

1

–1

–2

–3

–4

–5

A

B

Fig. II

Which figures you will use if measuring the length of OA and AB?

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

1b_ch8(1). 8.1rectangular coordinates a introduction to coordinate systems b rectangular coordinate...

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