-
Geometric figures
9Space and geometry
There are many different shapes that you see every day, in
buildings, on roads, in manufacturing and in artwork. Triangles and
quadrilaterals seem to be more common than circular shapes or other
polygons. This chapter looks at the language of geometry, geometric
properties and constructions involving triangles and
quadrilaterals.
-
recognise different types of polygons, including convex and
regular polygons
label and name points, intervals, equal intervals, equal angles,
lines, parallel lines, perpendicular lines, triangles and
quadrilaterals
recognise and classify triangles using sides and angles
recognise and classify quadrilaterals, including convex and
non-convex
quadrilaterals construct perpendicular lines and parallel lines
using set squares and
rulers construct various types of triangles and quadrilaterals
using compasses,
protractors, set squares and rulers investigate the properties
of triangles and quadrilaterals, including sides,
angles, diagonals, axes of symmetry, and order of rotational
symmetry.
polygon Any plane shape with straight sides. diagonal An
interval joining two non-adjacent vertices of a polygon. regular
polygon A polygon with all sides equal and all angles equal. convex
polygon A polygon whose vertices point outwards, not inwards,
where any interval joining two points on the polygon lies
completelyinside it.
interval Part of a line, with a starting point, an end point and
a definite length.
obtuse-angled triangle A triangle with one obtuse angle.
included angle The angle between two given sides of a polygon. set
square A ruling instrument in the shape of a right-angled
triangle.
Why is it impossible to construct a triangle with sides of
length 7 cm, 15 cm and 5 cm?
In this chapter you will:
Wordbank
Think!
GEOMETRIC FIGURES 283
CHAPTER 9
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284
NEW CENTURY MATHS 7
1 Draw:a a pair of perpendicular lines b a pair of parallel
lines.
2 Draw a rectangle and mark in any axes of symmetry.3 For the
quadrilateral shown on the right:
a name two intervals that are parallelb name two intervals that
are perpendicularc are the diagonals AC and DB equal in length?d
what is the size of DEC?e if DAB and ABC are cointerior and
DAB = 115, what is the size of ABC?4 a What type of triangle has
three equal sides?
b What type of quadrilateral has opposite sides parallel and all
angles measuring 90?5 Copy these shapes and mark in the axes of
symmetry on each one.
6 Which of the following shapes have rotational symmetry? State
the order of rotational symmetry for those that do.
7 For the rectangular prism shown on the right decide whether
each of the following is true (T) or false (F):a KL LP b NR II MQc
NM OP d KO II PQ.
8 a Draw a triangle that has an obtuse angle.b Draw a scalene
triangle.
C
A B
E
D
a b c d
e hf g
a b c
d e f
O
L
M
P
K
R Q
N
Start up
Worksheet 9-02
Symmetry
Worksheet 9-01
Brainstarters 9
Skillsheet 9-01
Line and rotational symmetry
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GEOMETRIC FIGURES
285
CHAPTER 9
Polygons
A polygon is a closed plane shape made up of straight sides. The
word polygon means many angles. These shapes are all polygons:
A polygon is named by the number of sides it has.
Convex and non-convex polygons
In Chapter 4, you learned about convex and non-convex solids. We
can also describe convex and non-convex polygons.Convex polygons
have vertices that point
outwards
while non-convex (or concave) polygons have vertices that point
or cave
inwards
.
Name Number of sides
Pentagon 5Hexagon 6Heptagon 7
Octagon 8Nonagon 9Decagon 10Undecagon 11Dodecagon 12
a b c d
e f g
h i j k l
SkillBuilder 23-01
Shapes I
A convex polygon A non-convex polygon
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286
NEW CENTURY MATHS 7
A simple test to determine if a polygon is convex or non-convex
is to draw any interval joining two points on the polygon, or any
diagonal joining two vertices of the polygon.
If the interval lies completely
inside
If all or part of the interval lies
outside
the polygon, then it is
convex
. of the polygon, then it is
non-convex
.
1 Name each of the polygons (a to l) on page 285, and state
which one is non-convex.2 How many sides has:
a a hexagon? b a quadrilateral? c a nonagon?d a decagon? e a
heptagon? f a pentagon?g a dodecagon? h an octagon? i an
undecagon?
3 Regular polygons have all sides equal and all angles equal.
Which of the polygons from Question 1 are regular?
4 Draw:a a regular hexagon b a non-regular hexagon c a regular
triangled a non-regular heptagon e a convex pentagon f a non-convex
dodecagon.
5 Which of the following shapes are not polygons?a trapezium b
ellipse c squared diamond e prism f circle
6 a Draw a pentagon with one axis of symmetry.b Draw a
quadrilateral with four axes of symmetryc Draw a hexagon with six
axes of symmetryd Draw a decagon with two axes of symmetry.
7 Geometry software, such as Cabri Geometry or Geometers
Sketchpad, can be used to demonstrate that you understand the words
you are required to learn in this chapter. Use this link to go to a
drawing exercise.
8 How many diagonals has a:a kite? b pentagon? c hexagon?
9 What shapes have been put together to form each of these
composite shapes?a b c
d e f
Exercise 9-01
Geometry 9-01
The vocabulary of geometry
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GEOMETRIC FIGURES 287 CHAPTER 9
10 Copy these composite shapes into your book and divide them
into the shapes requested.
11 Use Geometers Sketchpad or Cabri Geometry to create shapes
that a partner can try to draw using your instructions.
a b c
d e f g
Two pentagonsTwo triangles and one rectangle
One trapezium and one hexagon
Two trapeziums Four triangles One triangle and one trapezium
One square and one heptagon
Geometry 9-02
Creative copying
Working mathematically
Communicating: Logos and designs1 Find examples of company
logos. Draw them in your book. Discuss the shapes used to make
them.
2 Research some Islamic or Grecian art in the library or on the
Internet. Bring some pictures to class.
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288 NEW CENTURY MATHS 7
Classifying trianglesA polygon is two-dimensional (at) because
it has length and breadth (but not thickness). A triangle, having
three sides, is the simplest type of polygon. It is an important
shape that has been used throughout history and civilisations in
building, construction, packaging, and even as a cultural or
religious symbol.Triangles can be classied in two ways: by their
sides (equilateral, isosceles or scalene) by their angles
(acute-angled, obtuse-angled, right-angled).
Sides
EquilateralThree equal sides
Three equal angles
IsoscelesTwo equal sides
Two equal angles
ScaleneNo equal sides
No equal angles
Angles
Acute-angledAll three angles acute
Right-angledOne right angle
Obtuse-angledOne obtuse angle
Example 1Classify this triangle using sides and angles.
SolutionThis triangle has two equal sides and one obtuse
angle.It is isosceles and obtuse-angled.
R
ST
1 Rule up a table with these headings:
Place the letter of each of the following triangles under the
headings that match. (The same triangle may appear under more than
one heading.)
Acute-angled Obtuse-angled Right-angled Equilateral Isosceles
Scalene
Exercise 9-02
Worksheet 9-03
Properties of triangles
Example 1
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GEOMETRIC FIGURES 289 CHAPTER 9
2 Draw the following triangles:a a scalene triangle b a
right-angled scalene trianglec an isosceles triangle d an
equilateral trianglee a right-angled isosceles triangle f an acute
scalene triangle.
3 Use Geometers Sketchpad or Cabri Geometry to explore and draw
special trianglesand form your own denitions.
4 Is it possible to draw an equilateral right-angled triangle?
Why?5 Copy these triangles into your book and draw in all axes of
symmetry.
6 Do any triangles have rotational symmetry? Give examples to
support your answer.7 Is it possible to draw a triangle with two
obtuse angles? Why?8 The prex tri means three. Find the meaning of
each of the following mathematical
tri words:a trisect b trilateral c triangulate.
a b c
d e f
g h i
j k l
3088
62
140
20
3 cm
3 cm3 cm
12 cm
6 cm
15 cm
12 cm 12 cm
20
60
60 60
a b c
d e f
SkillBuilder 23-02
Shapes II
Geometry 9-03
Triangles
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290 NEW CENTURY MATHS 7
Naming geometric figuresPoints, lines and intervalsA point is a
position represented by a dot which is labelled by a capital
letter. The points on the right are labelled P, A and O.A line is a
straight edge that continues innitely (forever) in both directions,
so it is usually drawn with arrowheads at both ends. A line is
named by any two points on it. The line in this diagram is labelled
LM.An interval is a part of a line. It is a section of the line
with a starting point, an end point, and a denite length. An
interval is labelled by its two end-points. The interval in this
diagram is RS.
Reasoning and reecting: Building with shapes1 When you look at
the shapes of buildings and other constructions, you will
notice
that some shapes are more common than others. Write the names of
the most commonly used shapes.
2 When building any structure, strength is important. Which is
the strongest shape?
3 a Use ice block sticks or geo-sticks to make a triangle, a
square and a pentagon as shown above.
b Stand each shape up and push one corner. What happens?4 You
saw in Question 3 that a triangular framework is very strong or
rigid, which is
why that shape is used in many types of constructions. How can
you make the other shapes in Question 3 stronger?
5 Find as many pictures as you can of triangular frameworks in
everyday use. The ANZAC Bridge in Sydney is a good example.
Working mathematically
pointsP A
O
line
LM
intervalR S
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GEOMETRIC FIGURES 291 CHAPTER 9
TrianglesA triangle is identied by the capital letters that
label its vertices or angles. The triangle in this diagram is
labelled ABC or BCA or CAB (gures are usually labelled in clockwise
order).The angles of a triangle can be labelled by one letter or
three letters. The marked angle, shown, is T or RTS or STR.
The sides of a triangle can be described in two ways: by two
capital letters labelling their endpoints by a small letter that
matches the capital letter
naming its opposite angle.This diagram shows the triangle
ABC.The angles are labelled A, B, C.The sides are labelled a, b, c,
where side a is opposite A, side b is opposite B, and side c is
opposite C.The red side can be called CB, BC, or a.
QuadrilateralsA quadrilateral is any plane shape with four sides
and is identied by the capital letters that label its vertices or
angles. The quadrilateral in this diagram is labelled PQRS or QRSP
or RSPQ or SPQR.The angles of a quadrilateral can be labelled by
one letter or three letters. The marked angle in the quadrilateral
PQRS is R or QRS or SRQ.The sides of a quadrilateral can be
identied by the two capital letters labelling their endpoints. The
red side can be labelled PS or SP.
Equal angles and intervalsIn geometric diagrams, equal angles
are marked by identical symbols, while equal intervals are marked
by identical strokes. In this diagram, DF, DE and EG are all the
same length while F and DEF are the same size.
Parallel and perpendicular intervalsIn Chapter 2, we learned
about parallel and perpendicular lines. We can use the same
language and symbols to describe parallel and perpendicular
intervals.Parallel intervals point in the same direction and do not
intersect. In the rectangle WXYZ, shown, WZ is parallel to XY,
which is written WZ II XY.Perpendicular intervals meet at right
angles (90). In the rectangle WXYZ, WX is perpendicular to XY,
which is written as WX XY.
C B
A
S
R
TA
B
C
bc
aThe side labelled a is opposite the angle labelled A.
P
R
S
Q
D
F
EG
W X
Z Y
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292 NEW CENTURY MATHS 7
Example 2JKL is an isosceles triangle.The interval JM divides
JKL into two smaller triangles.a Name the two smaller triangles.b
What can be said about sides JL and JK?c Name the two equal angles
in JKL.d Explain the meaning of this sentence:
If JM LK, then LM = MK.Solutiona The two smaller triangles are
JML and JKM.b JL and JK have equal length.c JLM and JKM (or L and
K)d If side JM is perpendicular to side LK, then the lengths of
intervals LM and MK are equal.
a Draw a parallelogram and label it DEFG. b Mark both pairs of
parallel sides.c Name both pairs of parallel sides. d Mark the
equal angles D and F.e Mark the equal sides DG and EF.Solutiona The
answer should resemble the diagram on the right.b On the diagram,
one pair of parallel sides is marked by
arrows, and the other pair is marked by double arrows.c DE II GF
and DG II EF.d The equal angles D and F are marked by equal
arcs on the diagram.e The equal sides DG and EF are marked by
dashes on the diagram.
L
J
KM
Example 3
G
D
F
E
1 In this diagram, name a pair of intervals that are:a equalb
perpendicularc parallel.
2 a Draw two lines, PQ and RS, intersecting at T.b Mark the
equal angles PTR and STQ.c PTR = STQ. Why?
3 Copy the triangle KLM shown on the right, and correctly label
its sides k, l, and m.
G
D
FE
JIH
LM
K
Exercise 9-03Example 2
Example 3
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GEOMETRIC FIGURES 293 CHAPTER 9
4 What is the difference between the line EF and the interval
EF?
5 a Name the two triangles in the diagram on the right.b Name
the interval that is equal to:
i QR ii PT.c If TQRS is a trapezium, name the parallel sides.d
Copy the diagram and mark the parallel sides.e Mark the equal
angles PTQ and TSR.f PTQ = TSR. Why?
6 CDEF is a kite.a Copy the diagram and mark in the equal
sides.b What side is equal to DE?c Mark the equal angles F and D.d
Draw the two diagonals FD and CE.e Show on your diagram that FD
CE.
7 a Draw an isosceles triangle EFG where EF = EG.b Label the
sides of the triangle e, f, and g.c What is another name for the
side EF?d Mark on the triangle the equal angles F and G.
8 a Draw parallel lines AB and CD.b Draw a transversal EF
crossing both lines AB and CD, where EF AB.c CD EF. True or
false?
9 a What type of quadrilateral is STUV?b VS = ST. True or
false?c VS II ST. True or false?d = . True or false?e Name the
marked pair of equal sides.
10 a Draw a square, WXYZ, and mark all equal sides and angles.b
Name the point where side XY meets side ZY.c Name a pair of
parallel sides and mark them.d Name a pair of perpendicular sides.e
Explain the meaning of:
i WX XY ii WX = XY 11 a Draw a trapezium, UVWX, where UX =
VW.
b Mark the equal angles UXW and VWX.c UX II VW. True or false?d
UV II XW. True or false?
EF
S
T
P
R
Q
F
E
D
C
T
S
V
U
SV T VTU
W X
Z Y
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294 NEW CENTURY MATHS 7
Constructing trianglesTo construct a triangle we need to know
the length of its sides and the size of its angles. We also need a
ruler, a protractor and compasses.The following examples will show
you how to construct triangles. Hint: Draw a rough sketch before
beginning the construction.
Example 41 Construct a triangle with sides 3 cm,
5 cm and 4 cm.
SolutionStep 1: Draw an interval 5 cm long. (It is easier
to start with the longest side.)Step 2: Open the compasses to a
3 cm radius and
draw an arc from one end of the interval. (Every point on this
arc is 3 cm from the end of the interval.)
Step 3: Open the compasses to 4 cm and draw an arc from the
other end of the interval. (Every point on this second arc is 4 cm
from the other end of the interval.)
Step 4: Complete the triangle by joining the intersecting point
of the arcs to the ends of the interval.
2 Construct ABC where a = 5 cm, C = 30 and B = 70.
5 cm
3 cm 4 cmRough sketch
5 cm
5 cm3 c
m
5 cm
4 cm
4 cm
5 cm
3 cm
7030
a = 5 cm
A
B
C
bRough sketch c
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GEOMETRIC FIGURES 295 CHAPTER 9
SolutionStep 1: Draw an interval 5 cm long.
Step 2: Draw a 70 angle at B.Step 3: At C, draw a 30 angle.
Step 4: Join the arms to complete the triangle.
3 Construct a triangle with one side measuring 6 cm, another
side measuring 4 cm and an angle between them of 35.
SolutionStep 1: Draw an interval 6 cm long.
Step 2: Draw an angle of 35 at one end.
Step 3: Measure an interval of 4 cm onthe new arm.
Step 4: Complete the triangle.
Note: 35 is called the included angle because it is between the
two sides.
5 cm BC
70305 cm BC
70305 cm BC
A
4 cm
356 cm
Rough sketch
6 cm
356 cm
4 cm
356 cm
4 cm
356 cm
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296 NEW CENTURY MATHS 7
1 Construct each of these triangles accurately:
2 Geometers Sketchpad or Cabri Geometry can be used to
accurately draw triangles. This link will take you to an activity
that shows you how.
3 For each of the triangles constructed in Question 1:i name the
largest angle and the longest side
ii name the smallest angle and the shortest side.4 Copy and
complete this sentence:
In any triangle, the longest side is always the largest angle,
while the shortest side is always the angle.
5 a Which triangle in Question 1 is equilateral?b Measure its
angles. What do you notice?
6 a On a sheet of paper, construct an equilateral triangle of
side length 5 cm, and cut it out.b By folding along each of its
axes of symmetry, what do you observe about the sizes
of the triangles angles?c Measure the angles. What do you
notice?d Copy and complete:
An equilateral triangle has three sides, and three angles each
of size .7 Construct each of the following triangles. Draw a rough
sketch rst.
a ABC with a = 4 cm, b = 3 cm, and C = 50.b RST with r = 5 cm, s
= 3 cm, and t = 3 cm.c PQR with P = 60, Q = 60, and PQ = 4 cm.d LMN
with LN = 5 cm, ML = 4 cm, and NLM = 25.
8 a Which triangle in Question 7 is isosceles?b Measure its
angles. What do you notice?
9 a On a sheet of paper, construct an isosceles triangle with
two sides of length 6 cm, and cut it out.
b By folding, what do you observe about the sizes of the
triangles angles?c Measure the angles. What do you notice?d Copy
and complete:
An isosceles triangle has two sides, and two angles opposite
them.
a b c
d e f
4 cm
Q
S R
3 cm
4 cm40
M
O N6 cm80
C
E D3 cm
F
H G3 cm
3 cm3 cm
K
M L3 cm120
3 cm
X
Z
Y
3040
mm
130
2 cmExercise 9-04
Example 4
Geometry 9-04
Constructing triangles
Worksheet 9-03
Properties of triangles
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GEOMETRIC FIGURES 297 CHAPTER 9
Classifying quadrilateralsA quadrilateral is any shape with four
sides, but there are six special quadrilaterals that you need to
know. These are listed in the table below.
Name A quadrilateral with Diagrams
Trapezium one pair of opposite sides parallel
Parallelogram two pairs of opposite sides parallel
Rhombus(or diamond)
four equal sides
Rectangle four right angles
Square four equal sides and four right angles
Kite two pairs of adjacent sides equal
10 The triangle inequality rule says that if you add any two
sides of a triangle, the combined length is always greater than the
length of the third side. (This inequality can be written as a + b
c).a Test that this inequality is true for all of the triangles you
constructed in
Question 7.b Why is it impossible to construct a triangle with
sides of length 7 cm, 15 cm
and 5 cm?
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298 NEW CENTURY MATHS 7
Velcro sticksAll types of uses have been found for Velcro
fasteners in our world. They are used in the clothing industry, and
for attaching the chambers in artificial hearts, while astronauts
use them to fasten equipment so that it does not float away within
their space capsules.The idea for Velcro came to Georges de
Mestral, a Swiss engineer, in 1948. It is made of two surfaces, one
with hooks and one with loops. A thumbsize piece of Velcro contains
about 750 hooks and, on the other side, about 12 500 loops. De
Mestral conceived the idea of Velcro when he noticed tiny seed pods
caught in his socks after a walk in a forest.
Find four uses for Velcro.
Just for the record
Example 51 PQRS is a parallelogram, as shown on the right.
a Measure the lengths of its sides. Are opposite sides
equal?
b Measure the size of its angles. Are opposite angles equal?
c Does a parallelogram have line symmetry? If so, draw its axes
of symmetry.
d Does a parallelogram have rotational symmetry? If so, state
the order.e Draw the diagonals PR and QS and measure them. Are the
diagonals equal?Solutiona By measurement, PQ = SR = 4.2 cm and PS =
QR = 2 cm. Opposite sides are equal.b P = R = 100 and Q = S = 80.
Opposite angles are equal.c A parallelogram has no axes of
symmetry
(you cannot fold it in half).d A parallelogram has rotational
symmetry of
order 2. You can rotate it 180 so that it maps on to itself. The
centre of symmetry is marked O.
e PR = 4.2 cm while QS = 5 cm. The diagonals are not equal.
2 This diagram illustrates the properties of the diagonals of a
rectangle.a Are the diagonals equal?b Do the diagonals bisect each
other?c Do the diagonals intersect at right angles?d Do the
diagonals bisect the angles of the rectangle?
P Q
S R
P Q
S R
O
A B
D C
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GEOMETRIC FIGURES 299 CHAPTER 9
Solutiona The diagonals, AC and BD, have equal length.b The
diagonals bisect each other (cut each other in half), shown by the
equal markings.c The diagonals do not intersect at right angles.d
The diagonals do not bisect the angles of the rectangle, that is,
the right-angled
vertices (A, B, C, and D) are not cut into halves (45) by the
diagonals.
1 Find what the prex quad means. List other words beginning with
quad.2 Label each of these quadrilaterals as convex or
non-convex.
3 Make an enlarged copy of each of the quadrilaterals in the
diagram below or print out a copy. Cut out each quadrilateral and
name it.
a b c
d e f
Exercise 9-05
Worksheet9-04
Properties of quadrilaterals
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300 NEW CENTURY MATHS 7
4 Use Geometers Sketchpad or Cabri Geometry to explore the
quadrilaterals in Question 3 and form your own denitions.
5 a Copy or print out this table.
b Test the properties of each quadrilateral listed in the table
by folding and measuring them with a ruler, protractor and set
square. If the listed property is true, then place a tick in the
appropriate space. Write appropriate numbers in the last two
rows.
c Check your results with your teacher.d You should have noticed
that there are no ticks for the kite. Write two properties of
the
kite (that is two things that are always true about its sides,
angles or diagonals).6 Draw each of the following quadrilaterals
and mark all axes of symmetry.
a rectangle b square c parallelogramd rhombus e trapezium f
kite
7 List the quadrilaterals in Question 6 that have rotational
symmetry and mark the centre of symmetry, O, each time.
8 Use a ruler and protractor with the quadrilaterals you cut out
in Question 3, to discover the properties of the diagonals of each
one, as listed in the table below. Copy or print out this table.
Place ticks in the appropriate spaces.
9 Which quadrilateral am I? (There may be more than one
answer.)a My diagonals are equal. b All my sides are equal.c My
opposite sides are equal. d My diagonals bisect each other.e I have
four right angles. f I have two pairs of opposite sides parallel.g
I have rotational symmetry, but no axes of symmetry.h My diagonals
bisect each other at right angles.
10 a Does a square have the same properties as a rectangle? Why
do you think?b Does a rhombus have the same properties as a
parallelogram? Why do you think?
Trapezium Parallelogram Rhombus Rectangle Square Kite
Opposite sides are equal
Opposite sides are parallel
Opposite angles are equal
All angles are 90
Diagonals are equal
Number of axes of symmetry
Order of rotational symmetry
Trapezium Parallelogram Rhombus Rectangle Square Kite
Diagonals are equal
Diagonals bisect each other
Diagonals intersect at right angles
Diagonals bisect the angles of the quadrilateral
Example 5
Geometry 9-05
Quadrilaterals
SkillBuilder 23-11Axes of
symmetry
SkillBuilder 23-05
4-sided figures
Worksheet 9-04
Properties of quadrilaterals
Worksheet 9-04
Properties of quadrilaterals
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GEOMETRIC FIGURES 301 CHAPTER 9
Properties of triangles and quadrilaterals: a summaryShape
Properties
Equilateral triangle
All three sides equal All three angles 60 Three axes of
symmetry
Isosceles triangle
Two sides equal Two angles equal (opposite the equal sides) One
axis of symmetry
Scalene triangle
No sides or angles equal No axes of symmetry
Trapezium One pair of parallel sides No axes of symmetry
Kite Two pairs of adjacent sides equal One pair of opposite
angles equal One axis of symmetry Diagonals intersect at right
angles
Parallelogram Opposite sides equal and parallel Opposite angles
equal No axes of symmetry Diagonals bisect each other
Rhombus All four sides equal Opposite sides parallel Opposite
angles equal Two axes of symmetry Diagonals bisect at right angles
Diagonals bisect the angles of the rhombus
Rectangle All four angles 90 Opposite sides equal and parallel
Two axes of symmetry Diagonals are equal Diagonals bisect each
other
Square All four sides equal, all four angles 90 Four axes of
symmetry Diagonals are equal and bisect each other
at right angles Diagonals bisect the angles of the square
60 60
60
= 45
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302 NEW CENTURY MATHS 7
Applying strategies and reasoning: Shape puzzlesAll of the
shapes used in this activity can be printed out. Use the link to nd
them.
1 a How many squares can you nd in this shape? (The answer is
not 16!)
b How many rectangles can you nd?
2 Can you trace this shape without going over any line twice and
without lifting your pencil off the paper?
3 There are 12 different ways ve squares can be arranged. These
shapes are called pentominoes. The rst ve have been done for you.
Draw the other seven different arrangements.
4 Copy this equilateral triangle, cut out the four pieces and
rearrange them into a square.
5 How many triangles can you nd in each of these shapes?
AB
D
C
a b c d
Working mathematically
Worksheet 9-05
Shape puzzles
-
GEOMETRIC FIGURES 303 CHAPTER 9
Constructing perpendicular and parallel linesThe set squareA set
square is made in the shape of a right-angled triangle. It is used
for measuring and drawing right angles and for constructing
perpendicular and parallel lines. There are two types of set
squares, named according to their angle sizes:
Perpendicular lines and parallel lines can be constructed using
a set square or a protractor.
6 Copy this hexagon twice on to a piece of paper and then:a cut
the rst hexagon into two pieces
and rearrange them to make a parallelogram
b cut the second hexagon into three pieces and rearrange them to
make a rhombus.
60
30
45
45
The 6030 set square
The 45 set square
Example 6Use a set square to construct a line perpendicular to
XY through point W.
X
Y
W
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304 NEW CENTURY MATHS 7
Solution(A perpendicular line can also be constructed using a
protractor, by measuring a 90 angle.)
Use a set square to construct a line perpendicular to XY through
point Z (where Z is not on XY).
SolutionStep 1: Place the set square on the line XY.Step 2:
Place the ruler through the point Z.Step 3: Slide the set square
along XY until it
meets the ruler.Step 4: Slide the ruler until it ts the edge
of the set square and is perpendicular to XY.
Step 5: Rule the perpendicular line.
Use a set square to construct a line parallel to AB through
point P.
X
Y
W
Example 7
X
Y
Z
X
Y
Z
Example 8
P
A
B
-
GEOMETRIC FIGURES 305 CHAPTER 9
SolutionStep 1: Place the set square on the line AB.Step 2:
Place the ruler next to the set square,
at right angles to AB.Step 3: Hold the ruler rmly and slide the
set
square until its edge passes through point P.
Step 4: Rule the parallel line.(This construction can also be
done with a protractor replacing the set square, using the 90
mark.)
P
A
B
1 What is the name of the set square that is:a tall and thin?b
short and wide, and half of a square?c an isosceles, right-angled
triangle?d a scalene, right-angled triangle?
2 Why do you think the word square in set square is used to
describe a right angle? What other types of squares are used to
draw or measure right angles?
3 Draw a line and mark a point L on it. Construct a
perpendicular line through L.
4 Draw a line and mark a point X below it. Construct a
perpendicular line to the line through X.
5 Draw a line and mark a point P above it. Construct a parallel
line through P.
6 a Draw an interval, AB, 4 cm long. This will be the base of a
triangle.
b Mark X, the midpoint of AB, and construct a perpendicular
interval XC of length 5 cm at X.
c Join C to A and then to B to make a triangle, CBA.d What
special type of triangle is CBA?
7 a Draw a line XY and mark a point A above it. b Draw a line
parallel to XY through A.c Draw a line from A to XY perpendicular
to the line drawn in part b. Label the line AB.d What do you notice
about AB and XY?
L
X
P
X BA
C
Exercise 9-06
Example 6
Example 7
Example 8
-
306 NEW CENTURY MATHS 7
8 Are parallel lines always the same distance apart?a Draw a
pair of parallel lines and mark the
points D and E on one of them. b Draw perpendiculars from D and
E to the other
line. Where the lines intersect, mark the points F and G.
c Measure the lengths of DG and EF. What do you notice?
9 Draw two intervals that are parallel and of different lengths.
Join their ends to make a quadrilateral. What type of quadrilateral
have you constructed?
10 Draw an interval and mark its midpoint. Draw another interval
of a different length through this midpoint, perpendicular to the
rst interval. Join the ends of both intervals to make a
quadrilateral. What type of quadrilateral have you constructed?
11 Use the link to go to an exercise which uses Cabri Geometry
or Geometers Sketchpad to draw parallel and perpendicular
lines.
G
D
E
F
Geometry 9-06
Parallel and perpendicular
lines
Communicating and reecting: Vertical linesMany people use the
word perpendicular when they really mean vertical. A vertical line
is a line that is perpendicular to the Earths surface (or the
horizon). Vertical means up and down, while horizontal means at or
across.Brick walls are vertical.
Vertical lines are important when building homes, hanging
wallpaper and positioning pictures.1 Find out how builders use a
plumb bob to set out vertical lines.2 In your own words, write the
meanings of perpendicular and vertical.
Working mathematically
Right angle
-
GEOMETRIC FIGURES 307 CHAPTER 9
Constructing quadrilateralsExample 9Construct a square, FGHI, of
side length 4 cm.
SolutionStep 1: Construct the base, IH, of length 4 cm.Step 2:
Use a set square to construct the
perpendiculars, FI and GH, of length 4 cm.Step 3: Join FG.
Construct this kite, PQRS.
SolutionStep 1: Draw PS of length 3.5 cm.Step 2: Measure 105 at
S.Step 3: Construct SR of length 5 cm.Step 4: At R, use your
compass to draw an arc
of radius 5 cm.Step 5: At P, use your compass to draw an arc
of radius 3.5 cm.Step 6: The arcs cross at Q, the fourth vertex
of
the kite. Join P and R to Q.
Rough sketch
4 cmHI
GF
4 cm HI
GF
Example 10
P
R
QS
5 cm
3.5 cm
105
5 cm
P
R
QS
3.5 cm
105
3.5 cm
5 cm
Worksheet9-06
Constructions in diagrams
Worksheet9-08
Try drawing these!
Worksheet9-07
Constructionsin words
-
308 NEW CENTURY MATHS 7
1 a Draw BA measuring 6 cm. b Construct a perpendicular, BC, 3
cm long.c Complete the rectangle, ABCD.
2 Construct a square, KLMN, of side length 5 cm.
3 Construct this trapezium.
4 Construct a parallelogram with sides of 6 cm and 4 cm and an
included angle of 65.
5 a Construct two parallel intervals 4 cm long and 3 cm apart. b
Join the ends to make a quadrilateral. What type of
quadrilateral is it?c Measure the lengths of the two new sides.d
Are these sides both equal and parallel?
6 a Draw two joined intervals of the same length and use your
instruments to complete these shapes.
b What are these shapes called?
7 A trapezium with two equal (non-parallel) sides is called an
isosceles trapezium.a Name the equal sides in the isosceles
trapezium, PQRS, shown on the right. b Construct the isosceles
trapezium PQRS.c Measure all four angles of the trapezium.d Name
all pairs of equal angles.
8 Construct a rhombus with sides of 6 cm and an included angle
of 50.
6 cmB A
C
3 cm
6 cmU T
R
4 cm
S2 cm
60
3 cm
4 cm
4 cm
i ii
3 cm
3 cm
25 mm
25 mm
3 cmP Q
SR5 cm
4 cm4 cm
1 cm
Exercise 9-07
Example 9
-
GEOMETRIC FIGURES 309 CHAPTER 9
9 a Construct this quadrilateral. b What type of quadrilateral
is JKLM?
10 Construct this kite.
11 Construct the quadrilateral ABCD where AB BC, AB = 7 cm, BC =
3 cm, DC = 5 cm and AD = 4.5 cm.
12 Construct the trapezium DEFG where DE II GF, DE = 6 cm, EF =
GF = 3 cm and F = 135.
5.5 cmK100
J
M
3 cm
4 cm L80
Z X
Y
W
40
6 cm
4 cm
Example 10
1 Name each of the following polygons and state whether it is
convex or non-convex.
2 Explain the difference between parallel lines, perpendicular
lines and skew lines.3 Use the denitions of the quadrilaterals on
page 297 to help you answer these questions.
a Is the square a special type of rhombus?b Is the rhombus a
special type of square?c Is the parallelogram a special type of
trapezium?d Is the rectangle a special type of parallelogram?e Is
the parallelogram a special type of kite?f Is the rectangle a
special type of square?
a b c
d e f
Power plus
-
310 NEW CENTURY MATHS 7
4 a What additional property makes a parallelogram into a
rectangle? b What makes a kite into a rhombus?c What makes a
rectangle a square?
5 How many diagonals has:a a quadrilateral? b an octagon?c a
dodecagon?
6 Name all the quadrilaterals whose diagonals:a bisect each
other at right angles b bisect each otherc intersect at right
angles d have equal lengthe bisect the angles of the quadrilateral
f are equal and bisect each other.
7 a Draw an angle of any size, ABC.b Using only a ruler and
compasses, construct a rhombus
from ABC, with one vertex at B.c Bisect the angle ABC by drawing
one diagonal of the
rhombus.
8 Name the most general quadrilateral in which:a opposite angles
are equal b diagonals intersect at 90c diagonals are equal d all
angles are 90e opposite sides are parallel f diagonals bisect each
other.
9 a Construct a regular hexagon inside a circle of radius 5 cm.
b Construct a regular octagon inside a circle of radius 7 cm.
10 a Draw a line, AB, and a point, X, above it.
b Using only a ruler and compasses, construct a line through X
parallel to AB by creating a rhombus with one vertex at X and two
vertices on AB.
11 a Construct an interval AB and mark its midpoint M. b
Construct another interval CD perpendicular to AB through M, so
that M is also the
midpoint of CD.c Join the ends of both intervals to make a
quadrilateral. What type of quadrilateral have
you constructed?
12 a Repeat Question 11 but make sure that CD is the same length
as AB.b What type of quadrilateral have you constructed?
B C
A
X
AB
-
GEOMETRIC FIGURES 311 CHAPTER 9
Topic overview How useful do you think this chapter will be to
you in the future? Can you name any jobs which use some of the
concepts covered in this chapter? Did you have any problems with
any sections of this chapter? Discuss any problems with
a friend or your teacher.
Language of mathsacute-angled bisect compasses constructconvex
decagon diagonal equilateralincluded angle interval isosceles
kiteline symmetry obtuse-angled octagon orderparallel parallelogram
perpendicular polygonprotractor quadrilateral rectangle regular
polygonrhombus right-angled rotational symmetry scaleneset square
square trapezium vertex/vertices1 Draw a non-convex hexagon.2 What
is the difference between a line and an interval?3 The word
isosceles comes from Greece. Use a dictionary to nd out what it
means in Greek.4 What word in geometry means to cut in half?5
What is a set square and what is it used for?6 What is the more
common name for a regular quadrilateral?
Worksheet 9-09
Geometryfind-a-word
GEOMETRIC FIGURES
Quadrilaterals
Polygons
Naming geometric
figures
Constructing figures
TrianglesA ______ O______ R ______ E ______S ______ I ______
-
312 NEW CENTURY MATHS 7
Chapter 9 Review Topic testChapter 9
1 What type of polygon has 10 sides?
2 Name a shape that is not a polygon.
3 Draw:a a regular pentagon b a non-regular pentagonc a convex
quadrilateral d a non-convex quadrilateral.
4 Classify these triangles, by sides and angles.
5 a Classify FGH by sides and angles.b Which angles in FGH are
equal?
6 a Name a pair of parallel sides in this gure.b Name a pair of
perpendicular sides.c What type of quadrilateral is ABCD?
7 Construct the following triangles.a
b PQR with P = 20, PR = 3 cm and PQ = 4 cm.c MNO with MN = 4 cm,
NO = 5 cm and OM = 6 cm.
8 a Draw an obtuse-angled triangle, XYZ, and label its sides x,
y and z.b What is the relationship between the triangles longest
side and its largest angle?
Ex 9-01
Ex 9-01
Ex 9-01
Ex 9-02
a b c
d e f
Ex 9-02 F
H G
5 cm 4 cm
4 cm
Ex 9-03
C
B
D
A
Ex 9-04
A
C B
406 cm
40
Ex 9-04
-
GEOMETRIC FIGURES 313 CHAPTER 9
9 Name each of the following polygons:
10 a Copy each shape in Question 9 and mark all the axes of
symmetry.b List the shapes in Question 9 that have rotational
symmetry, and state the order
of rotational symmetry of each one.
11 What is the denition of a rhombus?
12 Write two properties of a parallelogram.
13 What polygon am I? (There may be more than one answer.)a I
have three sides and all of my angles are equal.b I am a
quadrilateral with opposite sides parallel.c I have ve sides.d I
have four sides and my diagonals bisect each other.e I am a
quadrilateral with one pair of parallel sides.f I have three sides.
My angles are 60, 80 and 40.
14 Copy this diagram and use a set square and ruler to construct
a line, through P:a perpendicular to QRb parallel to QR.
15 Construct this parallelogram.
16 Construct this quadrilateral.
Ex 9-05a b c
d e f
Ex 9-05
Ex 9-05
Ex 9-05
Ex 9-05
Ex 9-06Q
R
P
6 cm
4 cm
80
Ex 9-07
P5 cm
3 cm
Q
NM
55
4 cm
4 cm
Ex 9-07
Student textImprint pageTable of contentsPrefaceHow to use this
bookHow to use the CD-ROMAcknowledgementsSyllabus reference grid1
The history of numbersDifferent number systemsThe HinduArabic
number systemPlace valueExpanded notationThe four
operationsArithmagonsDividing by a two-digit numberOrder of
operationsThe symbols of mathematicsTopic overviewChapter
review
2 AnglesNaming anglesComparing angle sizeThe protractorDrawing
anglesAngle geometryNaming linesAngles and parallel linesFinding
parallel linesTopic overviewChapter review
3 Exploring numbersSpecial number patternsTests for
divisibilityFactorsPrime and composite numbersPrime factorsIndex
notationSquares, cubes and rootsTopic overviewChapter review
Mixed revision 14 SolidsNaming solidsConvex and non-convex
solidsPolyhedraPrisms and pyramidsCylinders, cones and
spheresClassifying solidsEulers ruleEdges of a solidThe Platonic
solidsDrawing and building solidsDifferent views of solidsTopic
overviewChapter review
5 IntegersNumber linesNumbers above and below zeroDirected
numbersOrdering directed numbersAdding and subtracting
integersMultiplying integersDividing integersThe four operations
with integersReading a map gridThe number planeThe number plane
with negative numbersTopic overviewChapter review
6 Patterns and rulesNumber rules from geometric patternsUsing
pattern rulesThe language of algebraTables of valuesFinding the
ruleFinding harder rulesFinding rules for geometric
patternsAlgebraic abbreviationsSubstitutionSubstitution with
negative numbersTopic overviewChapter review
Mixed revision 27 DecimalsPlace valueUnderstanding the
pointOrdering decimalsDecimals are special fractionsAdding and
subtracting decimalsMultiplying and dividing by powers of
10Multiplying decimalsCalculating changeDividing decimalsDecimals
at workConverting common fractions to decimalsRecurring
decimalsRounding decimalsMore decimals at workTopic overviewChapter
review
8 Length and areaThe history of measurementThe metric
systemConverting units of lengthReading measurement scalesThe
accuracy of measuring instrumentsEstimating and measuring
lengthPerimeterAreaConverting units of areaArea of squares,
rectangles and trianglesAreas of composite shapesMeasuring large
areasTopic overviewChapter review
9 Geometric figuresPolygonsClassifying trianglesNaming geometric
figuresConstructing trianglesClassifying quadrilateralsConstructing
perpendicular and parallel linesConstructing quadrilateralsTopic
overviewChapter review
Mixed revision 310 FractionsHighest common factor and lowest
common multipleNaming fractionsEquivalent fractionsOrdering
fractionsAdding and subtracting fractionsAdding and subtracting
mixed numeralsFractions of quantitiesMultiplying fractionsDividing
fractionsTopic overviewChapter review
11 Volume, mass and timeVolumeVolume of a rectangular
prismCapacity and liquid measureMassTimelinesConverting units of
timeTime calculationsWorld standard timesTimetablesTopic
overviewChapter review
12 AlgebraAlgebraic expressionsAlgebraic abbreviationsFrom words
to algebraic expressionsLike termsMultiplying algebraic
termsExpanding an expressionExpanding and simplifyingAlgebraic
substitutionTopic overviewChapter review
13 Interpreting graphs and tablesPicture graphsColumn graphs and
divided bar graphsSector graphsLine graphsTravel graphs and
conversion graphsStep graphsReading tablesTopic overviewChapter
review
Mixed revision 4General revisionAnswersIndex
GlossaryABCDEFG HI JK LMNOPQRSTU VW X Y Z
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