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Discovering
Geometry Workbook 1
For grades 7 - 9
L Hettiarachchi
DISCOVERING MATHEMATICS
Discovering Geometry Workbook 1
Part of the DISCOVERING MATHEMATICS series DISCOVERING MATHEMATICS
Acknowledgements I would like to thank the following people: My loving granddaughter Tara for her immense contribution in proof-reading this book. I am thrilled that you are so good at Mathematics. My beloved daughters and sons-in-law, each of whom have contributed in their own ways. My husband Aloysius for his encouragement over the years. Deepthi and Jude Samaranayake for all their support and practical help, especially with some of the drawings in this book. Your assistance helped me a great deal. The team at Action Station (www.action.sg) for their assistance with the design and layout of this book. All the students I have taught over my decades of teaching Mathematics, both in public schools and privately. You have taught me how to be a better teacher. Seeing you enjoy the subject, improve your grades and go on to be successful individuals in various professions has motivated me to continue teaching.
ISBN 978-624-96481-0-4
All right reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form without the prior written permission of the author. First published 2020
Dedicated with love to my darling granddaughter Alyssa Joy, who at the time of my writing this series is 3 years old. Thank you for bringing light to my life. I have no doubt that you will complete all the Mathematics workbooks someday without Grandma’s help! 1 August 2020.
Contents Chapter - 1
Introduction to Geometry
02- 22
Chapter - 4
Quadrilaterals
52 - 67
Basic geometrical terms………………....
02
Types of quadrilateral…………………….
52
Types of lines……………………………..
03
Sum of interior angles …………………..
57
Naming angles.........................................
05
Trapezium................................................
59
Classifying angles……………….………
06
Kite………………...………….…………...
60
Types of angles…..……………….………
07
Parallelogram….....………….……………
62
Measuring and drawing angles…..…….
08
Squares and rectangles………..………..
64
Complementary and Supplementary angles……...................………….............
11
Rhombus………………………………….
66
Adjacent angles on a straight line………
15
Mixed exercise…………………………....
67
Angles at a point…………………….……
18
Vertically opposite angles……………….
20
Mixed exercises…………………………..
22
Chapter - 2 Parallel Lines
4 - 37
Angles in parallel lines…………………...
24
Corresponding angles.……………….…..
28
Alternate angles.......................................
30
Co-interior angles………………………..
32
Mixed exercises..………………………….
34
Chapter - 3 Triangles
39 - 50
Chapter - 5 Polygons
69 - 82
Types of polygons……………….……….
69
Sum of interior angles.…………………...
72
Exterior and interior angles.....................
76
Number of sides……………...……..……
79
Mixed exercises..……………………….…
81
Chapter - 6
Geometric constructions
84 - 96
Basic geometrical constructions……......
84
Parallel lines……………………………....
86
Constructing angles of specific size........
87
Perpendicular bisectors………...……..…
88
Angle bisectors..……………….…………
89
Perpendicular from a point to a line…….
90
Types of triangles lines………………….
39
Sum of interior angles.…………………...
42
Isosceles and equilateral triangles..........
44
Perpendicular drawn to point on a line…...................…………......................
91
Sum of exterior angle……..……………...
47
Construction of triangles…………………
92
Mixed exercises..…………………………
49
Construction of quadrilaterals .………….
96
Chapter - 7
Pythagoras’ Theorem
98 - 108
Identifying hypotenuse…………………...
98
Writing an equation…………………..….
99
Calculating the hypotenuse ....................
100
Calculating a shorter side………………..
102
Mixed exercises..……………………….…
106
Chapter - 8 Mensuration
110 - 124
Formulae………………............................
110
Rectangles and squares…………………
111
Parallelograms.........................................
112
Rhombus……………….………………….
113
Trapezium…..……………….…………….
114
Triangle…..………………………………..
115
Kite……...................…………..................
116
Circle (area and circumference) ……..…
117
Length of an arc and area of a sector….
121
Mixed exercises…………………………..
124
Answers…………………………...........
125
Geometry is a very important part of Mathematics and it is essential for every student to have a strong foundation in the basic principles of Geometry. This Geometry workbook 1 is ideal as preparatory work for the students aiming to sit for GCE O’ level, IGCSE or GCSE examinations. It covers the Grade 7 to 9 Geometry syllabus of most curriculums. It is suitable for any student who wishes to learn Geometry from the basics and move progressively into more advanced topics. The workbook is packed with questions and drills to help build confidence to tackle different types of examinations questions. Geometry Workbook 1 is the first book of this series and it is followed by Geometry Workbook 2, and together they cover the entire Geometry syllabus for GCE O’ level or other exams at this level. This book is also suitable for any student wishing to work independently without the help of a teacher. To them I would say: Remember, ‘Practice makes perfect’. If you diligently complete the exercises and drills in both workbooks, I am very confident you will score high grades in your exams. It is my sincere hope that you will excel in Geometry and that you will not only score well but enjoy the subject and your learning experience. Good luck on your journey in learning Geometry. I welcome feedback from both teachers and students and can be reached by email at [email protected]. I would love to hear from you!
CHAPTER - 1 Introduction to Geometry
01
1
Introduction to Geometry
Geometry is the study of points, lines, angles, surfaces and solids.
Basic Geometrical Terms: Point A point is used to denote a specific location on a plane surface. We indicate the position of a point by placing a ‘•’ dot. A point has no size, no dimensions. Points are usually named by using an upper case single letter.
Line A line is a collection of points and extends to infinity in both directions. If a line is not straight, we usually refer to it as a curve or an arc. In plane Geometry, the word ‘line’ is usually taken to mean a straight line.
Line segment A line segment is a section of a line. It has two end points.
Ray A ray is a collection of points that begins at one point and extends to infnity in one direction
Angle Two rays with the same end points form an angle. Two rays are the arms of the angle formed. An angle is formed at A and AB and AC are its arms.
Vertex The common point B, where the two rays AB and AC meet, is called the vertex.
02
• A
1
Types of lines
Introduction to Geometry
Horizontal lines A horizontal line is one which runs from left to right across a page. It comes from the word ‘horizon’. The horizon is horizontal. Vertical lines
A vertical line runs from top to down on the page. A vertical line and a horizontal linecut at right angles.
Intersecting lines If two lines meet or cross at one point, they are intersecting. The point where they meet is called the point of intersection.
AB and CD are intersecting lines.
PQ and RS will only intersect when extended.
Parallel Lines Lines are parallel if they are the same distance apart over the entire length.
a
a
a
a
a
Perpendicular lines When two lines intersect to form a right angle, they are perpendicular lines. Symbol used to denote perpendicular lines.
03
is
1
Types of lines
Identify the following lines as parallel, perpendicular, horizontal, vertical or intersecting.
L5
L6
L4 a
a
a
a a a
a a
a
a
a
a
L3
a
a
a
a
a
a
a
a
L2
a
a
a
a
a
a
L1
L7
a
The lines L1 and L2 are ……………………………………………………………… a
a
a
a
a
a)
a
a
a
a
a
a
a
a a
a
a
b)
The lines L3 and L5 are …………………………………………………………………
c)
The lines L4 is
…………………………………………………………………
d)
The lines L3 is
…………………………………………………………………
e)
The lines L1 and L3 are …………………………………………………………………
f)
The lines L1 and L4 are ………………………………………………………………..
g)
The lines L4 and L5 are ………………………………………………………………...
h)
The lines L6 and L7 are ………………………………………………………………...
04
Introduction to Geometry
Exercise 1.1
1
Naming Angles
Introduction to Geometry
Angles can be named in various ways. The different ways of labelling an angle are shown below: P
● RQP
ꭓ⁰
●
● Q R
●ꭓ
a
a
a
Q
● PQR
●
a
Exercise 1.2
a
Name the following angles in four ways. b.
a
a
a
a.
c.
d.
T
---------------y
----------------
m⁰ ● a
k⁰
a
a
a
S
a
05 a
a
●
a
● R
----------------
X ●
z
----------------
1
Classifying Angles
>
Acute angle
90°
Obtuse angle > 90°
Right angle = 90°
Straight angle = 180°
Reflex angle > 180°
One complete turn = 360°
Exercise 1.3 Classify the following angles as Acute, Obtuse, Right, Straight or Reflex.
a.
(one
b.
----------------
---------------e.
d.
----------------
c.
----------------
---------------f.
----------------
06
g.
----------------
Introduction to Geometry
Angles can be classified as Acute, Obtuse, Right, Straight or Reflex according to the size of the given angle.
1
Type of Angles
Introduction to Geometry
Exercise 1.4 Identify the type of angles given below as acute, obtuse, right, straight or reflex. a)
b)
q a
b
c
p
a
r p ……………………..
a …………………….. a
q ……………………..
a
b …………………….. a c c)
r
…………………….. u
v
……………………..
m
d)
w
l
a
n a
l …………………
u ………………… ….. v ……………………..
m ……………………..
w ……………………..
n …………………
e)
f) q f
e
r
p
d
d
……………………..
e
……………………..
q …………………
f
……………………..
r …………………
p …………………….. a
a
a
07
a
1
Measuring and drawing angles
The size of the angle is based on how ‘widely open’ the angle is. The basic unit of measurement for an angle is the Degree. Angles can also be measured in Radians.
Size of angle = 90 ⁰
Introduction to Geometry
An angle measures the amount of turn. It is very simple to measure the size of the angles using a protractor.
Size of angle = 130⁰
Size of angle = 60 ⁰
Size of angle = 160 ⁰
---------------
08
---------------
1
Measuring Angles
Use a protractor to measure the following angles. a)
b) C •
•
•
B
A •
O
D
O ---------------
---------------
c)
d) F
E •
•
O E
• D •
O ---------------
---------------
e)
f) F
O
•
•
•
O
G
H
K
---------------
---------------
g)
•
h) O •
•
K •
M
• L
K
O ---------------
---------------
09
Introduction to Geometry
Exercise 1.5
1
Drawing angles
Use a protractor to draw the following angles given below. a)
PQR = 100⁰
c)
FOG = 72⁰
e)
EOF = 85⁰
f)
g)
WOX = 200⁰
LOM = 90⁰
10
Introduction to Geometry
Exercise 1.6
Adjacent, complementary and supplementary angles
Angles that share a vertex and a common side are called adjacent angles.
p
q
a
a b
a and b are adjacent angles a
a
a
a
a
p and q are adjacent angles
a
a
a
a
a
Complementary angles
If the sum of two angles is 90˚, then they are called complementary angles. One angle is the complement of the other. Angles do not have to be adjacent to be complementary.
50°
60° 30°
a
a
Supplementary angles
a
a
a a
a
a
a
a
a
40°
a
If the sum of the angles is 180˚, then they are called supplementary angles. One angle is the supplement of the other. The two angles do not have to be adjacent to be supplementary.
120°
40° 140°
60°
Angles are supplementary
a
a
Angles are supplementary a
a
a
a a
a
11
Introduction to Geometry
Adjacent angles
1
n
Complementary and Supplementary angles
i) Find the complement of each of the following angles.
a. 14˚ …………………….
b. 12˚ …………………….
c. 85˚ …………………….
d. 29˚ …………………….
e. 23˚ …………………….
f . 67˚ …………………….
g. 89˚ …………………….
h.53˚ …………………….
ii)
Find the supplement of each of the following angle.
iii)
State whether the following sets of angles are complementary, supplementary or neither. a) 69˚, 21˚ ………………………
b) 46˚, 134˚ ……………………...
c) 72˚,108˚ ……………………..
d) 153˚, 41˚ ……………………...
e) 96˚, 74˚ ……………………….
f) 26˚, 32˚ …………………..... .
g)111˚, 69 ˚ ……………………..
h) 102˚, 78˚ ……………………...
i) 88˚, 2 ˚ ……………………......
j) 90˚, 90˚ ……………………...
12
Introduction to Geometry
Exercise 1.7
1
1 Find the values of the unknown angles. a)
b)
c)
d)
g)
h)
13
Introduction to Geometry
Exercise 1.8
1 Find the values of the unknown angles.
21° ꭓ
y
32° 19°
31°
k h
39° 41°
29°
18°
b
28° m 31°
37° 28°
t
x 54° x
14
3t 2t
Introduction to Geometry
Exercise 1.9
1
Sum of adjacent angles
b
ꭓ
ꭓ
Introduction to Geometry
a
ꭓ ꭓ
4 ꭓ = 180°
a + b = 180°
Exercise 1.10 Find the unknown angles in the following diagrams.
y
4 ꭓ 3ꭓ 2 ꭓ
y 28° 58°
ꭓ
c)
d)
3t
n 30° n n
15
2t
t
1
Adjacent angles on a straight angle
Find the values of the unknown angles.
a)
b)
y
119°
122°
d)
c)
100° p
109°
m
f)
e) 46° t
39°
h)
g)
57°
106°
v
16
f
w
ꭓ
Introduction to Geometry
Exercise1.11
1
Supplementary angles
Introduction to Geometry
Exercise1.12 Find the values of the unknown angles.
a) b) y 35°
ꭓ
32° 44°
50° 70° t t
p 100° p
18° k k
k 66° k k
m
g)
h) 84° m m
v 2v
70 70
70 °
17
1
Angles at a point
Introduction to Geometry
The angles at a point add up to 360˚
a d
b c
a + b + c + d = 360˚
4 x 90˚ = 360˚
4m
Exercise 1.13
2m
Find the unknown angles in the following figures. b)
a)
240
8m 81° f
128˚
135˚
85° 168°
d
d a
a
a
a a
d)
c) 3t 5t
h 52°
t
18
46°
4m
1
Angles at a point
Find the values of the unknowns in the following diagrams.
240
a)
8m
b)
ꭓ
58°
108°
100°
p
c)
60°
3p q
d)
54° 3b d b
2m 4m
e)
3m
f)
v
2v
w
u
e f
d
d:e:f=1:3:5
19
Introduction to Geometry
Exercise1.14
1
Vertically Opposite Angles
b a
D
C
x y
30°
30° A
a = x (vertically opposite) b = y (vertically opposite)
O
AOC = BOD = 30˚ Not vertically opposite
Exercise1.15 Find the values of the unknown angles in the following diagrams. a)
b)
p q 115° r
p 38° r 72° s q
c)
d)
n 10°
q 100° p r s 42°
m
37° k
20
B
Introduction to Geometry
Vertically opposite angles are equal. When two straight lines intersect, the opposite angles are equal and are called vertically opposite angles.
1
Vertically opposite angles
Introduction to Geometry
Exercise1.16 Solve for the unknowns in the following diagrams. a)
b) 40°
60°
k
h
42° b
c)
c 2b
a
d)
d
e
f g
m m
n
32°
e)
f)
138° g
f h
b
h
a 147°
g)
h)
t- 10˚ t
x W
21
x +30° y
z
1
Mixed Exercises
Introduction to Geometry
Exercise 1.17 a) Identify a pair of adjacent angles in the following diagrams. i)
ii) b°
a°
a˚
b°
72°
72°
iv)
iii)
a°
a° b°
b°
Answer: ……………….. b) The angle A is double the size of its complement. Find the size of
.
c) Find the size of the angle which is 12˚ less than its complement.
d) The complement of an angle is
4 9
of its supplement. Find the size of the angle.
e) If the angles ꭓ + 6˚ and 2 ꭓ - 12˚ are supplementary, find their sizes.
f) Calculate the values of , y and z in the figure given below.
10Y+ 12°
ꭓ
z
15y - 28°
22
CHAPTER - 2 Parallel lines
23
2
Parallel lines
Angles in parallel lines Parallel lines are lines that are the same distance apart no matter how far they are extended. When a pair of parallel lines is cut by another straight line called a transversal, the angles formed have the following properties.
.
Co - interior angles
24
2
Angles in parallel lines
Classify the following angles as corresponding, alternate or co-interior angles.
j)
m)
k)
n)
25
l)
o)
Parallel lines
Exercise 2.1
2
Angles in parallel lines
Identify the following pairs of angles as corresponding, alternate or co-interior. a)
b)
c) a
p q
d)
e
b f
e)
g
f) f
h
k h
g)
j
h)
i) p
m
n
s q t
j)
k)
d
l) v
e
w
26
y
z
Parallel lines
Exercise 2.2
2
Angles in parallel lines
Insert the missing angles into the following diagrams to match the information given. b)
a)
c) g
p
y
y and z are alternate d)
g and h are corresponding e)
p and q are co-interior f)
e
a c o
20
a and b are cointerior g)
c and d are corresponding h)
r
i) j
u
j and k are co- interior
r and s are corresponding j)
k)
u and v are corresponding
l)
t
y y
w
t and u are alternate.
e and f are alternate
v and w are cointerior
27
o
20
y and z are alternate
Parallel lines
Exercise 2.3
2
Corresponding angles
Parallel lines
Exercise 2.4 v VIdentify the pairs of corresponding angles in the following diagrams.
d)
e)
f)
h)
i)
28
2
Corresponding angles
Parallel lines
Exercise 2.5 v V Find the unknown angles in the following diagrams.
119° q
138° q
z y
f
m n
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
121°
a
a
e
a
m 141°
78° m
t
p
s
n 28°
a
a
a
a
a
a
a
a
a
a
a
q
a
a
a
a
a
a a
a
a
a
a
a
a
a
a
a a
u t
a
47° m
a
a
z
h k
y
a
a
143° a
a
a
110°
a
29
a
a
a
a
a
a
a
a
a
a
a a
a
a
a
a
a
a
a
a
a
a
a
a a
a
a a
26˚
2
Alternate angles
Parallel lines
Exercise 2.6 V
Identify alternate angles in the following diagrams. a)
b)
c)
k
j
b c
e
g
f
m e d a a
a
a
a
w
a
d e g
y n
q
p
a
a
a
30
a
a
a
a
a
a a
a
a
a
a
a
a
a
a
a
z
f
a
v u
a
a a
a
a
a
a
a
a
a
a
a
a
a
a a
aa
a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
i
h
2
Alternate angles
Parallel lines
Exercise 2.7 V
Find the values of the unknown angles in the following diagrams.
e
p t
g
f
h 40°
a
a
a
a
a
a
a a
a
a a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a a
a
a
a
a
a
a
a
42°
143°
q
a
a
122°
t
52°
a
a
m
a
62°
28° 43°
f
a
v a
a
a
a
a
a
31
a
a
a
a
a
a
a
a
a
a a
a
a
a
48°
h
g
m
a
t u
a
p q
a
a
a
a
a
a
a
h k
a
a
a
a
a
110°
au
a
a
a
a
a
a
p
a
a
a
a
a a
a
a
a
a
73°
a
a
a
80°
n
k
a
a
f
a
m
2
Co-Interior angles
Parallel lines
Exercise 2.8 V
Identify co-interior angles in the following diagrams. a)
b)
c)
f j g
f
e
f
g
a
a
a
a
a a
a
a a
a
a
a
a
a
a
a
a
a
a
a
af)
a
a
e)
a
d)
a
a
a
a
a a
a
a
j
d
k l
a
p q r s
f g h i a
a b
a a
a a
a a
a
a
a
a aa
a
a
a a
aa
a
a
a
a
aa
a
a
a
32
a
a
a
a
a
a
a
a
a a
a
a
a
a a
a
a
a
a
a
a
aa
a
a
a
a
aa
a a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a a a a
a
a
aa
a
a
a a a
a
a
a
a
a
a
a
a
f
u v w mo q
q s f g q i hq q p a
a
g
t
h
aa
k s
p
a
r
a
l
a
a
a
a
a
a
a
g)
a a
a
a
a a a
a
a
a
a
a
a
a
a
a a
a
a
a
aa
a
a
a
a
a a
a
a
a
k a
r
j
m
a
q
a
l
a
p
h
i
2
Co- interior angles
Parallel lines
Exercise 2.9
V
Find the values of the unknown angles in the following diagrams.
p
132°
72°
a
a
a
a
a
a
a a
a
a
a
a
a
a
a
a
a
a
a
a
a
m
a
125°
a
b
a
a
a
m
v
108°
a
a
a
w
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
74°
132 °
a
a
a
a
a
v v v q
z 122°
112°
q v v 105° q
a a
a
a
a
a
a
33
a
a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
vvvvq
2
Mixed Exercises
Parallel lines
Exercise 2.10 Find the values of the unknown angles in the following diagrams.
a)
b)
c)
y z u
145°
d)
m
r 112°
156°
e)
f)
149° h
k
p
n 2n
72°
z
y
2y 72°
98°
m
Z
i)
k 120°
35°
a
92° h
n m
34
120°
72° b c
2
Mixed Exercises
Parallel lines
Exercise 2.11 Find the values of the unknown angles in the following diagrams. a)
b)
a
c)
c y
3h
u
b d 122°
h
2u
k
d)
f)
e)
120°
g
56° k
p
g)
i 53°
m m
h)
i)
35
2
Mixed Exercises
Parallel lines
Exercise 2.12 Find the unknown angles in the following diagrams.
f g
47°
f
150°
h 72°
h
k
22°
118° f
y
30°
40°
42°
104° f f 32° g
32°
42°
t h
140°
36
h
2
Mixed Exercises
Parallel lines
Exercise 2.13 Find the unknown angles in the following diagrams.
210° f m
23°
118° x
42° t
18°
22°
155° f
18° 83°
165° m
58°
z
4w 3w
t
258°
2w
38°
37
w
CHAPTER - 3 Triangles
38
3
Triangles
Types of Triangles A triangle is a closed plane figure bounded by three line segments. It is the smallest member of the polygon family.
Triangles can be classified according to their sides.
60˚
60˚
60˚
Equilateral triangle
Isosceles triangle
Scalene triangle
Triangles can also be classified according to their angles.
˂90˚
˂90˚
>90˚
Right – angled triangle
Obtuse – angled triangle
39
˂90˚
Acute – angled triangle
3
Types of Triangles
a.
b.
a a
7.5cm
5.2 cm
6.8cm
6cm 9.2cm
5.2 cm
c.
------------------------------
------------------------------
a
a
7 cm
d.
a a
9 cm
10 cm
a
7cm
a
14 cm
7 cm
------------------------------
------------------------------
a
a
e.
f. a
7 cm 4 cm 5 cm ------------------------------
------------------------------
a
a
g.
h.
------------------------------
-----------------------------a
40
a
Triangles
Exercise 3.1 Identify each of the following triangles as scalene, isosceles or equilateral.
3
Types of Triangles Classify each of the following as an acute, obtuse or right - angled triangle. a.
b.
42°
60° 102°
38°
c. a
18 o
_____________ __ o
_____________ ___
d.
20
52
a
88°
a
o o
97
o
72
31o _____________
______________
e.
f.
6 cm
6 cm
6 cm
65
o
58°
6 cm
a
a a
57
o
_______________ ______
_______________ _
g.
h.
115 33
o
o
32
o
50°
_____________ o
88
64
i.
j. 32
o
84°
Clas sify eac h of the foll owi ng as acu 41te,
70°
o
22
o
Triangles
Exercise 3.2
3
Sum of interior angles
Exercise 3.3 Find the unknown angles in the following figures. a.
b. y
a
a
d
57o 100o
76o
48o
d. c. a
p
40o
a
81° 42°
f.
p
e.
a
a
a o
34
52°
123°
a o
30
k
i.
o
h. 53o
a
50
t
94o
g.
32°
j.
w
a
68°
a
40
o
42
f
56°
Triangles
The sum of the interior angles of a triangle is .180˚
3
Sum of Interior angles
Triangles
Exercise 3.4 Find the unknown angles in the following diagrams. a.
b.
a
a
f
3f
y
2y 2y
80°
c.
d. p
a
a
w 2w
100°
108°
3p
e.
f.
2g
b
a
a
b
2g g g.
h.
a
a
3t
m 88° 2t 4t
3m
43
3
Isosceles Triangles
p
p
Equilateral Triangles 60° 60°
60°
All three sides of an equilateral triangle are equal. All three angles of an equilateral triangle are equal and each angle is 60˚.
Exercise 3.5 Find the values of the unknown angles in the following diagrams. a. b.
32°
c.
p
65°
m
b 70°
a
d.
q
n
f.
e.
a
2m
2a
y
a
a
z
n
b
g. a
i.
h.
a
b
32°
n
b
a
c
5a
2a
44
p
a
2q 3r
Triangles
Angles opposite to equal sides of an isosceles triangle are also equal. The sides opposite to the equal angles of this triangle are also equal.
q
3
Exterior angles of a triangle
Triangles
When each side of a triangle is extended three exterior angles are formed. The sides can be extended either clockwise or anti-clockwise. b+c
a An exterior angle of a triangle is equal to the sum of two interior opposite angles.
b
c
a+b
a+c Exercise 3.6 Find the unknown angles in the following diagrams.
a.
b.
c. 84°
42°
z
y
76°
d.
64°
52°
q
32°
e.
f.
38° 48°
m 140°
91˚ t
57° 62° d
45
3
Exterior angles
Triangles
Exercise 3.7 Find the values of the unknowns in the following figures. a.
b.
33°
p
c.
z
143° 164°
118°
41°
e.
d.
110°
f.
84°
138°
z
m
143°
n
122° y
g.
h.
i. a
a a
a
w
b 2a
b
32°
46
3
The Sum of Exterior Angles
Triangles
The sum of the exterior angles of a triangle is 360°
Exercise 3.8
Find the unknown angles in the following diagrams. a.
b.
112°
102°
a vv
a
m
131°
c.
154°
d.
143°
v
y
124°
n
111°
109°
k
e. v
a
3f
4f
f.
b
98°
m
2f
n
c
z g.
2y
h.
c
v
b
a
n
m
2
a:b:c=2:3:4
47
y
2y
3
Sum of Exterior Angles
Triangles
Exercise 3.9 Find the angles given by unknowns. a.
b.
a
a
2y
4e 2e
3y
4e
4v
c.
d. 3w
a
x
a
v
v
70°
2w
y
z
y:z=3:4
f.
e.
me
d
a v d
h
n
54°
e:m=2:3
g.
m
h. m
48°
b
3p
a
4p
48
3
Mixed Exercises
Triangles
Exercise 3.10 Find the unknown angles in the following figures. a.
b.
t
36
154°
72°
o
m
38° 168°
d.
c.
t
h
22°
s
46° 50°
123°
33° 62°
t
e.
44°
vv
108°
52°
44°
f.
66°
h
w
d
g.
h h
y
h.
32°
h
24°
k
h
78°
58°
40°
49
3
Mixed Exercises
Triangles
Exercise 3.11 Find the unknown angles in the following figures. a.
b.
p
v
vv
118°
h
h
y
52° h
50° 42°
c.
x x
70°
d.
h
3x
120°
p
a
w
e.
b
42° x
f.
v
y
110°
g.
h.
40°
k
c
28° 115°
58° h
b a
m h
50
38°
CHAPTER - 4 Quadrilaterals
51
4
Quadrilaterals
Types of quadrilaterals A quadrilateral is a polygon with four sides and four vertices. The sum of four interior angles of any quadrilateral is 360 Types of special quadrilaterals: Trapezium Properties: One pair of opposite sides is parallel.
An isosceles trapezium has non-parallel sides equal.
Parallelogram
Properties: i) Opposite sides are equal. ii) The diagonals bisect each other. iii) Opposite angles are equal. iv) Each diagonal separates it into two congruent triangles.
Rectangle
A rectangle is a parallelogram and has all the properties of a parallelogram. In addition, it has the following properties: i) Each interior angle is 90˚. ii) Diagonals are equal.
52
4
Types of quadrilaterals
Quadrilaterals
Rhombus
A rhombus is a parallelogram and has all the properties of a parallelogram. In addition, it has the following properties: i) All four sides are equal. ii) The diagonal bisect at right angles. iii) The diagonals bisect the angles at the vertices.
Square
A square is a parallelogram and has all the properties of a parallelogram. In addition, i) All four angles are right angle. ii) Diagonals are equal. iii) Diagonals bisect at right angles. iv) Diagonals bisect angles at the vertices and each bisected angle is 45˚.
Kite Properties: i) It consists of two isosceles triangles on the opposite sides of the same base. ii) The longer diagonal is the line of symmetry of the kite. iii) The longer diagonal bisects the shorter one at right angles and bisects the angles at the vertices.
53
4
Types of quadrilaterals
Quadrilaterals
The above description is illustrated in the diagram below:
Quadrilateral
Kite Parallelogram
Rhombus
Rectangle
Square
54
Trapezium
Isosceles Trapezium
4
Types of quadrilaterals
Classify each of the following figures as a quadrilateral, trapezium, isosceles trapezium, kite, parallelogram, rectangle, rhombus or square. a.
b.
c.
˃
˃
------------------
d.
------------------
------------------
e.
˃
f.
˃ ------------------
g.
------------------
h.
------------------
------------------
i.
------------------
------------------
j.
------------------
------------------
55
------------------
Quadrilaterals
Exercise 4.1
4
Types of quadrilaterals
State whether each of the following is true or false. a. A trapezium is a quadrilateral with exactly one pair of parallel sides. ............................... b. A parallelogram is a trapezium. ............................... c. A trapezium is a parallelogram. ............................... d. A rhombus is a square. ............................... e. A square is a rhombus. ............................... f.
A kite is a rhombus. ...............................
g. A rhombus is a kite. ............................... h. An isosceles trapezium is a parallelogram. ............................... i.
A parallelogram is an isosceles trapezium. ...............................
j.
A parallelogram is a rhombus. ...............................
k. A rhombus is a parallelogram. ............................... l.
Any kite is a quadrilateral. ...............................
m. Any quadrilateral is a kite. ............................... n. A kite has perpendicular diagonals. ............................... o. A parallelogram has only one pair of parallel sides. ............................... p. A parallelogram has two equal diagonals. ............................... q. If quadrilateral is a square, then it is also a rectangle. ........................ r.
If a quadrilateral is a rectangle, then it is also a square. ............................... Exercise 4.3
a. List two ways a parallelogram and a rhombus are different. ............................................................................................................................................. b. How many pairs of equal sides does an isosceles trapezium have?................................... c. Name two quadrilaterals that have diagonals bisecting each other at right angles. ............................................................................................................................................. d. Name a quadrilateral that has diagonals making 45˚ at the vertices?................................ e. How many pairs of equal sides does a kite have?............................................................
56
Quadrilaterals
Exercise 4.2
4
Sum of interior angles
Find the values of the missing angles in the following quadrilaterals. a.
b.
42° 42°
53°
c.
d. 92°
114°
56° 100°
e.
f. 104° 99°
102°
88°
66°
g.
h.
108° 98° 83°
70° 88°
57
Quadrilaterals
Exercise 4.4
4
Sum of interior angles
Quadrilaterals
Exercise 4.5 Find the value of ‘x’ in each of the following quadrilaterals. a.
b. 4
30˚
3
2
2
c.
3
d.
5
6
145˚ 2 3
e.
3
4
3
f.
2
96˚
3
2
4
g.
h. 2
3
3 3 +40˚ 152˚
58
4
Trapezium
Quadrilaterals
Exercise 4.6 Find the unknown angles in the following trapeziums. a.
b. 82° 112°
48°
84°
c.
d. 146° 53° 78°
e.
f. 41°
62°
z 46°
g.
h.
59° 68°
59
47°
4
Kite
Quadrilaterals
Exercise 4.7 Find the unknown angles in the following kites. a.
b. 58°
22°
78°
52°
c.
d. m 22° n
70° 34°
68°
e.
f.
68°
56° 38° 92°
g.
h. 46°
32°
120° 31°
60
4
Kite
Quadrilaterals
Exercise 4.8 Find the unknown angles in the following kites. a.
b. 84° 65°
55°
48°
mn
c.
d. 36°
58°
24° z 124°
e.
f. 100° 36°
24° z
48°
g.
h. 28° 39°
102°
61
24°
4
Parallelogram
Quadrilaterals
Exercise 4.9 Find the unknown angles in the following parallelograms. a.
b. 108 ° 52°
c.
d. 61°
48 °
e.
f. 88°
64°
128°
g.
h. 118° 32
°
103° 21 °
62
4
Parallelogram
Quadrilaterals
Exercise 4.10 (i) Find the unknown values of each side of the following parallelograms. a.
2
b.
−3
5
3 +4 +4
c.
=
=
Y=
d.
3y - 1
2 +5
3 −4
+5
=
= y=
(ii) Find in cm, the length of each diagonal of the following parallelograms. a.
b.
A
P
S D
B
AC=
PR =
+ 10
C
BD=
Q SQ =
R (iii) Find in cm, the value of each side of the parallelograms. B
a.
b. P
A
Q
3 −2
C D
AB =
S
BC =
10 Y+2
PQ =
R
QR =
AD =
PS =
DC =
SR =
63
4
Squares and Rectangles
a. ABCD is a square. AB = 5 cm .
b. ABCD is a square. AC = 7.1 cm. Find BD
Find i) BC ii) CD iii) AD A
B
D
C
c. ABCD is a square. AX = 3.5 cm. Find
i) BX
ii) CX
A
A
B
D
C
d. ABCD is a square.
iii) DX
Find A
B
i) a ii) b iii) c
b°
iv) d v) e
a°
X
e°
B
d° c°
D
D
C
64
C
Quadrilaterals
Exercise 4.11
4
Squares and Rectangles
a. ABCD is a square. Find i) a˚ ii) b˚
b. PQRS is a rectangle. PQ=8cm, QR = 6cm, PR =10cm.
iii)c˚
Find i) RS ii) PS iii) QS A
B
b°
Q
P
a°
D
C
c°
d. PQRS is a rectangle. ∠ PXQ = 1180
c. PQRS is a rectangle. PX = 5 cm. Find
i) RX
ii) QS
P
R
S
Find the unknowns a,b,c,d,e,f,g and h. P
Q
X S
S
65
b
Q
X
f
R
a
h
g
e
d
c
R
Quadrilaterals
Exercise 4.12
4
Rhombus
Quadrilaterals
Exercise 4.13 Find the values of x and y in the following rhombuses. a.
b.
c.
68˚
y
52˚
2y
=
=
y=
y=
d.
60˚
4y = 2x
y=
= y=
e.
f. Given that
:
= 3: 4
10
=
=
y=
y=
66
4
Mixed Exercises
Quadrilaterals
Exercise 4.14 i) In the figure, PS = PT and PT \\ QR. Find the values of q and r. P q
r
Q 142 ˚
72˚ T
S
R
ii) In the following figure, ABCD is a parallelogram. Given that BE = BC, find x, y and z. z
A
B ꭓ
D
Y
48˚
C
E
iii) Given that PQRS is a parallelogram and QRT is an equilateral triangle, find x and y. P
Q
ꭓ
T y
2ꭓ
S
R
iv) If ABCD and ABDE are parallelograms, find the value of y. B
A Y˚
118 ˚ D
E
67
26˚
C
CHAPTER - 5 Polygons
68
5
Polygons
Types of polygons A polygon is a closed figure bounded by three or more straight line-segments and is classified according to its number of sides. Poly means many and gon means sides. Thus a polygon has many sides.
convex
concave
A polygon can be either convex or concave. A convex polygon has each interior angle less than 180° . A polygon that has one or more interior angle greater than 180° is a concave polygon. Many of the basic polygons that we learn about in Geometry are convex polygons.
Regular and irregular polygons.
A regular polygon has all its sides of equal length and all its angles of equal measures.
An irregular polygon does not have all its sides or angles equal.
69
5
Types of Polygons
Sum of No of No of sides/angles triangles interior angles (n) Si = 180˚(n-2) (n-2)
Figure
Triangle 3
1
180˚
Quadrilateral 4
2
360˚
5
3
540˚
6
4
720˚
Pentagon
Hexagon
70
Polygons
Name
5
Types of Polygons Figure
No of No of Sum of sides/Angles Triangles interior angles (n)
(n-2)
S i = 180˚ (n-2)
Heptagon
7
5
900˚
8
6
1080˚
10
8
1440˚
12
10
1800˚
Octagon
Decagon
Dodecagon
71
Polygons
Name
5
Sum of interior angles
a. 14 sides
b. 18 sides
c. 22 sides
d. 30 sides
e. 36 sides
f. 40 sides
g. 44 sides
h. 60 sides
72
Polygons
Exercise 5.1 Use Si = 180˚ (n-2) to find the sum of interior angles of the following polygons with a given number of sides.
5
Sum of interior angles
Polygons
Exercise 5.2 Find the unknown values in the following polygons. a.
b. 123°
102° 169°
p
112°
98°
a
160°
158˚ 164°
81°
110°
169°
c.
d. 148°
172° 115°
95°
141°
130° t
132° 124°
206°
78°
f
106°
w
e.
145°
f.
168 °
170°
174° 169°
171° 105°
145°
169°
170° d
89°
173° 168 °
73
158°
5
Interior angles
Find the size of each interior angle of the following regular polygons with a) 15 sides
b) 72 sides
Si = 180˚( 15 – 2 )
Si = 180˚( 72– 2 )
180 x 13
180 x 70
Size of one angle = 1940 15
Size of one angle = 12600 72
= 156˚
= 175˚
Exercise 5.3 Find the size of each interior angle of the following regular polygons. a. 18 sides
b. 24 sides
c. 45 sides
d. 48 sides
e. 56 sides
f. 60 sides
74
Polygons
Example :
5
Number of sides of a polygon
Find the number of (i) triangles (ii) sides of each of the polygons, where the sum of interior angles ( Si ) is given.
a.
Si = 1260°
b. S i = 720°
c. S i = 2340°
d. S i = 1980°
e. S i = 4680°
f. S i = 3420°
g. S i = 2700°
h. S i = 6120°
i. Si
75
Polygons
Exercise 5.4
5
Exterior angles
Polygons
The sum of the exterior angles of a polygon is 360°. Exercise 5.5
Find the unknown values in each of the following irregular polygons. a.
b. 68°
48°
53°
t
52°
48°
28°
k
58°
80° 72° c.
28°
d. W°
63° p
38°
96°
68°
33° 44° 32°
48°
38°
e.
f.
62°
84°
78°
64°
47°
44°
52° 73°
y
50°
43°
44°
v g.
h. o
64°
40 85˚
p
z
p 49°
o
41
46° 58
o
o
73
58°
76
88°
5
Exterior and interior angles Exercise 5.6 Find the size of an exterior angle of each of the the following polygons where the size of an interior angle is given. a. 108°
b. 162°
c. 168°
d. 150°
e. 165°
f. 156°
Exercise 5.7 Find the size of an interior angle of each of the following polygons where the size of an exterior angle is given.
a. 68°
b. 49°
c. 84°
d. 15°
e. 65°
f. 56°
77
Polygons
Interior and exterior angles of a polygon are supplementary.
5
Exterior and interior angles
Find the size of (i) an exterior angle (ii) an interior angle of each of the following regular polygons. Hint: first find the size of an exterior angle and then the interior angle. a. Pentagon
b. Octagon
c. Enneagon
d. Heptagon
e. Hexagon
f. Decagon
g. Dodecagon
h. Hex decagon
i. Octdecagon
j. Icosagon
78
Polygons
Exercise 5.8
5
Number of sides
i) Find the number of sides of each of the following regular polygons, where the size of an exterior angle is giv en.
a. 30°
b. 12°
c. 24°
d. 15°
e. 36°
f. 18°
g. 72˚
h. 06˚
79
Polygons
Exercise 5.9
5
Number of sides
Find the number of sides of the following regular polygons where the size of an interior. angle is given. Hint: find the exterior angle first.
a. 144°
b. 162°
c. 165°
d. 150°
e. 135°
f. 156°
g. 168°
h. 170°
80
Polygons
Exercise 5.10
5
M ixed Exercise
a. The diagram below consists of two regular hexagons. Two extended sides meet at A. Find x. o
x
o
A
b. In the following figure, ABCDE is a regular polygon. Find the values of h and k. A B E C D c. Given that PQRSTU is a regular hexagon, find the values of w and z . P U
Q
T
R S
d. ABCDE and BFGEC are two regular pentagons. Find the values of p, q and r. F B
G
A
p°
E
r°
q° E
C
D
81
Polygons
Exercise 5.11
5
Mixed Exercise
Polygons
Exercise 5.12
P T U
xo
Q S
R
a. In the figure above, PQRST is a regular pentagon and PT and RS are extended to meet at U. Calculate the value of x.˚
b. In the figure given below, PQRST is a regular pentagon and PXYR is a square. Calculate the value of .˚ P
X T
Q S
Y
˚
R
82
CHAPTER - 6 Geometric Constructions
83
6
Geometric Constructions
The following sketches are for you to understand the basic constructions which you should know. 1. Using set squares to draw a parallel line to a given line through a given point. The figure below illustrates how to use a ruler and set squares to draw a parallel line to AB.
2. Perpendicular bisector of line XY
3. Perpendicular from P to AB
P
X
Y A
4. Perpendicular raised at point P
P
84
B
6
Geometric Constructions
o
ABC
6. 60 angle at point D
A
B
C
D
7. 90o angle at point E
E
85
Geometric Constructions
5. Bisector of
6
Parallel lines
Use a ruler and set squares to draw a parallel line to the given line through the given point.
a)
b) ●
●
c)
d) ●
●
e)
f)
●
●
●
●
86
Geometric Constructions
Exercise 6.1
6
Constructing angles
Geometric Constructions
Exercise 6.2 Construct the following angles without using a protractor. a) 60 o
b) 30o
o
d) 45
c) 90
o
o
f) 135
e) 15
o
o
o
g) 120
h) 105
i) 75
j) 150o
87
6
Perpendicular bisector
Construct the perpendicular bisector of each of the following lines. a)
b)
c)
d)
e)
f)
g)
h)
88
Geometric Constructions
Exercise 6.3
6
Angle bisector
Construct the bisector of each of the following angles.
a)
b)
c) d)
e)
f)
g) h)
89
Geometric Constructions
Exercise 6.4
6
Perpendicular from a point to a line
Construct a perpendicular for each of the following lines from the points given.
A
B
C
D
E
F
g) h)
G
H
90
Geometric Constructions
Exercise 6.5
6
Perpendicular drawn to a point on the line
Geometric Constructions
Exercise 6.6 For each of the line shown below, draw the perpendicular at P. a)
b)
P P
c)
d) P
P P
e)
f)
P P
g)
h)
P P
P
91
6
Construction of triangles
Construct the following triangles (SSS). Use an additional she et of paper if necessary. a) Δ ABC in which AB=5cm, BC=6.5cm, AC=5.8cm. b) Δ PQR in which PQ=4.8cm, QR=5.2cm, PR=4.8cm. c) Δ LMN in which PQ=6.0cm, QR=5.9cm, PR =7.0cm. d) Δ DEF in which DE=6.2cm, EF=7.0cm, DF=6.0cm.
92
Geometric Constructions
Exercise 6.7
6
Construction of triangles
Construct the following triangles (SAS). Use an additional sheet of paper if necessary. a) ∆ LMN in which LM=5cm, LN=6.2cm,
MLN=80o
b) ∆ PQR in which PQ=4.8cm, QR=7.0cm, PQR=110o c) ∆ DEF in which DE=6.0cm, DF=5.0cm, EDF=100o d) ∆ BAC in which BA=4.0cm, BC=7.2cm, ABC=72 o
93
Geometric Constructions
Exercise 6.8
6
Construction of triangles
Construct the following triangles (ASA). Use an additional sheet of paper if necessary. FGH=45,o
a)
FGH in which GH=7.2cm,
b)
DEF in which EF=6.8cm,
DEF=51,o
c)
PQR in which PR=6.8cm,
QPR=40,o
d)
KLM in which LM=6.7cm,
KLM=80,o
GHF=50o EFD=43o PQR=80o LKM=64o
94
Geometric Constructions
Exercise 6.9
6
Construction of triangles
Construct the following triangles (RHS). Use an additional sheet of paper if necessary. a) ∆ RST in which ST=4.2cm,
o
RST=90 , RT=8.2cm
b) ∆ QPR in which PR=4.8cm, QPR=90,o QR=8.6cm c) ∆ ABC in which AB=4.6cm,
CAB=90 o, BC=8.0cm
d) ∆ LMN in which MN=5.6cm,
LMN=90o, EF=7.2 cm
95
Geometric Constructions
Exercise 6.10
6
Construction of quadrilaterals
Construct the following polygons. Use an additional sheet of paper if necessary. a) Rectangle ABCD, given that AB=4.7cm, BC=6.2cm. b) Rectangle PQRS, given that PQ=7.1cm, PS=4.3cm. o
c) Parallelogram BCDE, given that BC=8.0cm, CBE=50 , BC=5.0cm. o
d) Parallelogram CDEF, given that CD=7.0cm, CDE=120, DE=5.2cm . e) Parallelogram PQRS, given that PQ=4.6cm,
96
o
PQR=110, QR=7.2cm
Geometric Constructions
Exercise 6.11
CHAPTER - 7 Pythagoras’ Theorem
97
7
7
Pythagoras’ Theorem
A
B Pythagoras, a Greek Mathematician, discovered that for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. C A Therefore, Pythagoras’ theorem states that in ABC,
B B
2
BC = AB 2+ AC2
C
Identifying hypotenuse
D
E
Exercise 7.1 Identify the hypotenuse of each of the following triangles. a)
B A
PQ
C
b)
F D
R
c)
E
L T M
…………
S Q
A L
d)
R
………… Z
e) U
N
M
X
T
Y
…………
f) Y
X
S
N
…………
U
98
…………
F
Z
…………
Pythagoras’ Theorem
The longest side of any right- angled triangle is opposite its right angle. It is known as the ‘hypotenuse’.
7
Writing an equation Use Pythagoras’ Theorem to write an equation connecting the three sides of the following right-angled triangles. a)
b) a d
c
e
b
…………………………
c)
…………………………
d)
p q
………………………… e)
………………………… f)
u
………………………
99
…………………………
Pythagoras’ Theorem
Exercise 7.2
7
Calculating the hypotenuse
Find the value of
Pythagoras’ Theorem
Exercise 7.3 in each of the following right -angled triangles.
a)
b)
6
8
c)
d)
e)
f) 7
24
100
7
Calculating the hypotenuse
Find the value of a)
in the following right-angled triangles. b)
16 12
15
20
c)
d) y
y
13
84
e)
f) 126
32 y y
101
Pythagoras’ Theorem
Exercise 7.4
7
Finding a shorter side
Pythagoras’ Theorem
Exercise 7.5 Apply Pythagoras’ theorem to find the value of z in the following figures. a)
b) z
12
z
20
c)
d) z
z
35
34
30
21
e)
f) 24
z
36 40 39
z
102
7
Finding a shorter side
Find the value of the unknown in each of the following diagrams. All dimensions in cm. a)
b)
60.
p
52
48
61
c)
d)
44
r
20
55 29 q
e)
f)
w
117 84
k
85 45
103
Pythagoras’ Theorem
Exercise 7.6
7
Finding a shorter side
Pythagoras’ Theorem
Exercise 7.7 Find the unknown side in each of the following triangles. All dimensions in cm. a)
b)
65
36
85
63
c)
d) m
74
24 74
70
e)
f)
42
169 58
120
104
7
Mixed Exercises
Find the unknown values in each of the following triangles, giving your answers to 2 decimal places. All dimensions in cm.
a)
b)
y
80.3
3.5 60.3
4.5
c)
x
d)
2.8
8.7
6.5 p
z
12.4
11.5
e)
t
f)
60.2
d
105
4.5
Pythagoras’ Theorem
Exercise 7.8
7
Mixed Exercises
Find the length of the side marked with a letter in each of the following triangles.
a)
b)
5.2 cm
u t
8.7 cm
8.9
cm
4.2 cm
c)
d) 6.3 cm
x
12.7 cm
y
8 6.
cm
e)
f) 42
cm
7.
.5
8
m
28.6 cm
cm
n
11.3 cm
106
Pythagoras’ Theorem
Exercise 7.9
7
Mixed Exercises
Pythagoras’ Theorem
Exercise 7.10 Find the unknown values in the following diagrams. All dimensions in cm. a)
b) S
A P 15
12
9
5
B
5
C
4
D
R
Q
Find the length of
Find the length of
i) AC
i) QR
ii) AD
ii) PR
c)
d)
H
D
10 A
17
G E
7 25
9
C
B
12
F
Find the length of
Find the length of
i) EG
i) AC
ii) HG
ii) AD
107
7
Mixed Exercises
Pythagoras’ Theorem
Exercise 7.11 Find the unknown values in the following diagrams. All dimensions in cm.
A
a)
b)
21
B
E
24
F
40
28 D
12
C
G
60
Find the length of
Find the length of
i) AC
i) FG
ii) AD
ii) FH
c)
H
d) x 6
8
y
8
y
5
14
y
Find the value of y in the above diagram,
Find the values of x and y in the
giving the answer correct to 2 decimal places.
above diagram, giving the answer correct to 2 decimal places.
108
CHAPTER - 8 Mensuration
109
8
Mensuration
Perimeter and area of plane figures Perimeter is the total length of the outline of a shape and area is the extent of the shape enclosed within the perimeter.
Rectangle = 2 +2 Area = lb
= 4
Square: =
Parallelogram
h1
h
= 2 + 2
d
Area = bh or dh 1
b
= 4 Rhombs
Area =
h
(d1 X d 2 ) or bh
b = ( a +b + c + d )
a c
d
h
Area =
h ( a + b)
b
Triangle
h
c
a
a
h
= ( a +b + c )
c Area =
b
b 110
bh
8
Rectangles and Squares
Mensuration
Exercise 8.1 (i) Find the perimeter and area of the following figures. b)
5 cm
c)
2 cm
5 cm
2 cm
3cm 5 cm
a)
2 cm
3 cm
2 cm
e)
x
d)
x 5 cm
3 cm
3 cm
4 cm
1cm
2 cm x
x 1cm
1cm
6 cm
(ii) Find the perimeter and the area of the shaded region for each of the following figures. a)
b)
6m
20cm 111
14cm
16cm
12cm
8
Parallelogram
Mensuration
Exercise 8.2 Find the perimeter and the area of the following parellelograms. a)
b) 4m
5m 8m
c) Find the value of b in the parallelogram.
Given that the area of the following parallelogram is 80 cm 2, find its perimeter.
5m
6 cm
e) Given that b : d = 2 : 1 and the area of the parallelogram is 84 cm2, find its perimeter.
d b
f)
Use the following figure to fill in the blanks in the table. H2
B1 12 6 8
B2 H1 B1
21
112
B2 6 7
H1 4
H2
Area 84 cm 2
15 12
10 30 7
240 cm 2 189 cm 2
8
Rhombus
a) Use the following rhombus to fill in the blanks in the given table. b 10 5 9
i) ii) iii) iv)
h
h 6 5 9
d1 12 10 9
d2 8 9 6
b b) The area of the rhombus in the figure is 72 cm2. If AX = 6 cm, find DX. A B D
C c) Given that the shaded area of the following rhombus is 48 cm 2 and BX = 4 cm, find AX. B A C
d) In the following rhombus, PX = 5cm, SR = 12cm and OP = 3cm. Find QS. P
Q
O
5
S
X
12
R
113
Mensuration
Exercise 8.3
8
Trapezium
a) In the trapezium given below, a = 9 m, b = 12 m and h = 8 m. Find the area of the trapezium. a
h
b
b) Use the diagram below to fill in the given table.
h b
a
b
h
8 cm
12 cm
5 cm
18 cm
9 cm
14 cm
8 cm
P
C
Q
S U
88 cm2
D
A
B
d)
108 cm2 110 cm2
8 cm
9 cm
c) Given that AD = 12 m, BC = 10 m and the area of trapezium is 77 m 2, find DC.
Area
R
Y
In the figure, the area of the trapezium is twice the area of the rectangle. PQ = 6 cm, QR = 2 cm and RY = 3cm. Find UT.
T A
B
e) If AB : CD = 3 : 4, BX = 10 cm and area of the trapezium is 105 cm2, find AB and CD. D
114
X
C
Mensuration
Exercise 8.4
8
Triangle
Mensuration
Exercise 8.5 a) Use the following figure to complete the table.
i) ii) iii) iv)
h b
b 5 cm 7 cm 15 cm
h Area 8 cm 12 cm 9 cm 72 cm2 90 cm2
b) Use the following figure to complete the table.
i) ii) iii) iv)
h b
b 6 cm 4 cm 17 cm
h Area 9 cm 13 cm 10 cm 60 cm2 51 cm2
c) Use the following figure to complete the table.
i) ii) h1
b1
b1 h1 b2 h2 Area 5 cm 12 cm 3 cm 8 cm 12 cm 4 cm
iii) 16 cm 8 cm 40 cm2 iv) 10 cm 15 cm 60 cm2 v
8 cm
18 cm
36 cm2
d) The ratio of the base to its height of a given triangle is 5 : 2. If the area of the triangle is 20 cm2, find the lengths of the base and the height of the triangle.
115
8
Kite
X
d1 A
d2
Area of a kite = C
(d 1 X d 2 ) where ‘d 1’
and ‘d 2 ’ are the two diagonals.
D Exercise 8.6 Use the above figure and answer the following questions on kites. a) Given that AC = 12 cm and BD = 8 cm , find the area of ABCD.
b) The area of kite ABCD is 36 cm,2 where BX = 3 cm and AX : CX = 2 : 1. Find the length of BC.
c) It is given that the area of kite ABCD is 144 cm2 and BD : AC = 1 : 2. Find DX.
d) Given that AD = 15 cm and BD = 18 cm, calculate the area of the kite ABCD.
e) If AC : BD = 9 : 4 and the area of ABCD is 288 cm2, find the length of each diagonal.
f) In the figure, AX : CX = 3 : 2, BX = 3 cm and area of triangle BXC is 6cm2. Find the area of the kite ABCD.
116
Mensuration
B
8
Circumference and area of a circle
Mensuration
Circumference of a circle (C) = 2 Area of a circle = centre
•
. 14
radius
= 1 2 1 2
Exercise 8.7
= Type equation here.
Using = a.
, find the circumference and the area of the following circles. b.
.
.
c.
d.
.
.
5.6 cm f.
e.
.
.
117
21cm
Circumference and area
Example 1 : Find the (i) diameter (ii)area of a circle if its circumference is 40 cm. C = 2 πr 40 = 2πr r= = 6.37 cm ( 2 dec. pl.) d = 2 x 6.37 = 12.74 cm (calculator display) = A= (6.37) 2 = 127.48 cm2 ( 2 dec. pl.) 2
Example 2 : Given that the area of a circle is 176.7 cm , find the (i) diameter (ii) circumference of the circle. = 176.7 = . r2 = = 56.24 r = √56.24 = 7.50 cm ( 2 dec. pl.) Diameter = 15.0 cm Circumference = d = 47.12cm Exercise 8.8 a) Complete the following table. You may use a calculator to get the vaue of answers correct to two decimal places. r (i) (ii)
d
Circumference Area (A) (C)
5.2 cm 46.5 cm 46.8 cm 2
(iii) (iv) •
(r)
, giving
12.6 cm
(v)
96.4 cm 214.8 cm2
(vi) (vii)
145.6 cm
(viii)
18.2 cm
118
Mensuration
8
8
Quadrants and Semicircles
Perimeter = = 1
+ 2
3 4
Quadrant
Area = Perimeter =
+2
Area = Perimeter =
Exercise 8.9 Find the area and the perimeter of the following figures ( = a) b)
).
10.5cm 14cm
d)
28 cm
50 cm
c)
14 cm 21 cm
119
+2
Mensuration
Semicircle
8
8
8
Composite figures
Mensuration
Exercise 8.10
Find the area and the perimeter of the following figures. a)
b)
c) 16cm
7cm
25cm
6 cm
28cm
16cm
d)
e)
12 cm
5cm
14 cm
24cm
f) Find the area and perimeter of the shaded region. 35cm 28cm
120
3.5cm
8
Length of Arc and Area of Sector
minor arc ABC
A
Length of an arc =
C minor sector O
Area of a sector =
D Exercise 8.11 Find the length of the minor arc and the area of the minor sector ( =
c)
˚
x2 x πr
where ˚ is the angle subtended by the arc at the centre.
major sector major arc ADC
a)
˚
b)
d)
120˚
121
).
Mensuration
B
8
Length of an arc and Area of a sector
Taking
=
, find the (i) length of minor arc (ii) the area of the minor sector.
a)
b)
o
160
c)
d)
63cm
e)
f)
o
60
g)
h)
o
54
122
Mensuration
Exercise 8.12
8
Length of arc and Area of Sector
Taking
=
, find the (i) length of the major arc (ii) the area of the major sector.
a)
b)
o 200 .
. 80o
c)
d)
.
.
e)
f)
.
.
150˚
48o
123
Mensuration
Exercise 8.13
8
Mixed Exercises
a) Complete the following table Radius Angle at the center i
14cm
135°
ii
4.2m
90°
iii
31.5cm
200°
iv
8.4m
40°
v
10.5cm
72°
vi
28cm
144°
vii
21.0cm
300°
viii
120mm
120°
ix
4.9m
100°
x
140mm
215°
xi
11.2m
150°
xii
0.7m
45°
xiii
91mm
108°
xiv
14.7cm
168°
xv
70m
225°
Arc length
Area of sector
Mensuration
Exercise 8.14
b) In the diagram given below, ABC is a semicircle and AC is the diameter. AD = CD = 13 cm and AC = 10 cm. Taking = 3.14, find (i) the perimeter (ii) the total area of the figure.
B
10cm
A
D
C
124
125
126
127
128
129
polygons
130
131
132
133
134
This Geometry Workbook is ideal as preparatory work for the students aiming to sit for GCE O’ level, IGCSE or GCSE examinations and covers Years 8 to 10 Geometry
It is also suitable for any student wishing to work independently
DISCOVERING MATHEMATICS
Other books available in the DISCOVERING MATHEMATICS series include: Discovering Geometry - Workbook 2 Discovering Algebra and Graphs - Workbook 1 Discovering Algebra and Graph - Workbook 2 Discovering Arithmetic - Workbook 1 Discovering Arithmetic - Workbook 2 Discovering Statistics and Probability - Workbook Discovering Matrics and Transformation - Workbook Discovering Sets, Functions and Vectors - Workbook Discovering Trigonometry - Workbook