1A_Ch61

1
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2
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6.1 Basic Geometric Knowledge
Index
3
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6.2 Plane Figures

• Introduction toPlane Figures

Index
4
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6.3 Three-dimensional Figures
Index
5
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6.4 Polyhedra
Index
6
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6.1 Basic Geometric Knowledge
Points, Lines and Planes
A)
1. Refer to the right figure.
i. A is a point. ii. BE is a line. iii. CD is a
line segment,C and D are calledthe end points
of that line segment. iv. Figure ACD represents a
plane.
Index
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6.1 Basic Geometric Knowledge

• Example

Points, Lines and Planes
A)
2. Relations among Points, Lines and
Planes i. The straight line in the common part
of two planes is called the line of
intersection. ii. The two lines meet each other
at a point, that point is called the point of
intersection.
Index

• Index 6.1

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6.1 Basic Geometric Knowledge

• Name all the line segments and planes in the
  given figure.
• Which point is the point of intersection of MQ
  and PN ?

(a) Line segments
MN, MO, NO, OP, OQ, PQ, MQ, NP
Planes
MNO, OPQ
(b) Point of intersection of MQ and PN
O

• Key Concept 6.1.1

Index
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6.1 Basic Geometric Knowledge

• Example

Types of Angles
B)
Angles can be classified according to their
sizes as follows
Index

• Index 6.1

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6.1 Basic Geometric Knowledge
What kind of angle is each of the following
angles in the given figure? (a) ?AOB (b) ?BOD
(a) ?AOB 180
? ?AOB is a straight angle.
(b) ?BOD ?BOC ?COD 140 90 50
? ?BOD is an acute angle.
Index
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6.1 Basic Geometric Knowledge
What kind of angle is each of the following
angles in the given figure?
(a) ?AFE (b) ?AHD (c) ?EFB
(a) ?AFE 120 ? ?AFE is an obtuse angle.
(b) ?AHD 90 ? ?AHD is a right angle.
(c) ?EFB ?EFA ?AFB 120 60 180
? ?EFB is a straight angle.

• Key Concept 6.1.2

Index
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6.1 Basic Geometric Knowledge

• Example

Parallel and Perpendicular Lines
C)
i. RS and TU are a pair of parallel lines. We can
write RS // TU. ii. AB and TU are a pair of
perpendicular lines. We can write AB ? TU.
iii. Parallel and perpendicular lines can be
constructed by a ruler and a set square.
Index

• Index 6.1

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6.1 Basic Geometric Knowledge
Name all the parallel lines and perpendicular
lines in the given figure.
Parallel lines
AG // DC // FE, GF // DE
AB ? BC, AG ? GF, GF ? FE, FE ? ED, ED ? DC
Perpendicular lines

• Key Concept 6.1.3

Index
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6.2 Plane Figures
Introduction to Plane Figures
1. A geometric figure formed by points, lines and
planes lying in the same plane is called a plane
figure.
E.g.
Index

• Index 6.2

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6.2 Plane Figures
Circles
A)

• 1. O is the centre.
• 2. OP is the radius.
• 3. AOB is the diameter.
• 4. The curve AQBPA which forms the entire circle
  isthe circumference.
• The curve AP is part of the circumference, called
  an arc of the circle.
• 6. Circles and arcs can be constructed by a pair
  of compasses.

Index
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6.2 Plane Figures

• Example

Circles
A)
Note i. Circumference, radius and
diameter can represent lengths as
well. ii. Diameter 2 radius
Index

• Index 6.2

17
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6.2 Plane Figures
It is known that O is the centre of each of the
following circles, find the values of the
unknowns. (a) (b)
(a) Diameter 12 cm
(b) Radius 4.5 m
? x 12 2 6
? y 4.5 2 9

• Key Concept 6.2.2

Index
18
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6.2 Plane Figures

• Example

Triangles
B)

• In the above triangle,

i. the line segments AB, BC and CA are called the
sides of ?ABC, ii. points A, B, C are called the
vertices (singular vertex) of ?ABC.
2. The sum of the three angles of a triangle is
180.
Index
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6.2 Plane Figures
Triangles
B)
3. Classification of triangles
Index
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6.2 Plane Figures

• Example

Triangles
B)
3. Classification of triangles
Index
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6.2 Plane Figures

• Example

Triangles
B)

• Triangles can be constructed by a protractor and
  a pair of compasses etc. according to given
  conditions
• i. Given three sides of a triangle.
• ii. Given two sides and the included angle of a
  triangle.

Index

• Index 6.2

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6.2 Plane Figures
Find the unknown angle a in the figure.
a 120 40 180
a 160 180
a 180 160
20
Index
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6.2 Plane Figures
Find the unknowns x and y in ?ABC as shown.
In ?ABD,
In ?ABC,
x 62 90 180
28 62 48 y 180
x 152 180
138 y 180
x 180 152
y 180 138
28
42

• Key Concept 6.2.3

Index
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6.2 Plane Figures

• For the above triangles A, B, C and D, identify
• scalene obtuse-angled triangle?
• isosceles acute-angled triangle?

(a) C (b) D

• Key Concept 6.2.4

Index
25
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6.2 Plane Figures
Construct ?ABC, where AB 4 cm, BC 3 cm and AC
3.5 cm.
1. Use a ruler to draw a line segment AB of 4
length cm. 2. With centre at A and radius 3.5 cm,
use a pair of compasses to draw an arc. 3. With
centre at B and radius 3 cm, use a pair of
compasses to draw another arc. 4. The two arcs
drawn should meet at C. 5. Join AC, then BC. ?ABC
is drawn.
Index
26
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6.2 Plane Figures
Construct ?PQR, where PQ 3 cm, ?RPQ 50 and
RP 4 cm.
1. Use a ruler to draw a line segment PQ of
length 3 cm. 2. Use a protractor to draw ?TPQ
that measures 50. 3. Use a ruler to mark a point
R on PT produced such that RP 4 cm. 4. Join QR,
then ?PQR is drawn.

• Key Concept 6.2.6

Index
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6.2 Plane Figures
Polygons
C)

• A plane figure formed by 3 or more line segments
  is called a polygon.
• A polygon is usually named by the number of its
  sides or n-sided polygon (n is whole number).

Index
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6.2 Plane Figures
Polygons
C)

• The line segments that form a polygon are called
  sides of the polygon.
• The point where two adjacent sides meet is called
  a vertex of the polygon.
• The line segment joining two non-adjacent
  vertices is called a diagonal.

Index
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6.2 Plane Figures

• Example

Polygons
C)
Classification of polygons
Index

• Index 6.2

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6.2 Plane Figures
For each of the following polygons, state whether
it is (a) an equilateral polygon (b)
an equiangular polygon (c) a regular polygon.
(a) A, C (b) A, B (c) A

• Key Concept 6.2.8

Index
31
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6.3 Three-dimensional Figures
Introduction
A)
1. A solid is an object that occupies space.
2. The surfaces of a solid are called
faces. 3. The line segment on a solid that is
formed by any two intersecting faces is called an
edge. 4. A point that is formed by 3 or more
intersecting faces on a solid is called a vertex.
Index

• Index 6.3

32
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6.3 Three-dimensional Figures

• Example

Sketch the Two-dimensional (2-D) Representation
of Simple Solids
B)
1. We can use solid and dotted lines to draw
rough 2-D figures of solids on a plane.
2. We can also use isometric drawings to draw
more accurate 2-D figures of solids on a plane.
Index
33
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6.3 Three-dimensional Figures

• Example

Sketch the Two-dimensional (2-D) Representation
of Simple Solids
B)
The face obtained by cutting a solid along a
certain plane is called a cross-section of the
solid. If we cut the solid at different
positions, we may obtain different cross-sections.
Note If we obtain the same cross-sections by
cutting a solid along certain direction, then the
cross-sections are called uniform cross-sections.
Index

• Index 6.3

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6.3 Three-dimensional Figures
Use an isometric dotted paper to draw the 2-D
representation of the box.
Index
35
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6.3 Three-dimensional Figures
Use an isometric grid paper to draw the 2-D
representation of the given solid.

• Key Concept 6.3.2

Index
36
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6.3 Three-dimensional Figures
Which of the following faces represents the
cross-section of the given solid when it is cut
vertically along the blue line?
The cross-section is B.
Index
37
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6.3 Three-dimensional Figures
Draw the cross-section of the given solid when it
is cut horizontally along the yellow line.

• Key Concept 6.3.3

Index
38
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6.4 Polyhedra

• Example

Introduction to Polyhedra
A)
If all the faces of a solid are polygons, then
that solid is called a polyhedron.
Note The polyhedra can be named by their
numbers of faces.
Index

• Index 6.4

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6.4 Polyhedra
Determine which of the following solids is not a
polyhedron.
B

• Key Concept 6.4.1

Index
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6.4 Polyhedra

• Example

Making Models of Polyhedra
B)
We can use a net to make a model of polyhedron.
Index

• Index 6.4

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6.4 Polyhedra
The diagram on the right is a polyhedron. Which
of the following net do you think can make that
polyhedron?
A
B

• Key Concept 6.4.2

A
Index