Chapter09 Geometric Figures

9

Space and geometry

Geometric figures

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.

In this chapter you will: ■ ■ ■ ■ ■ ■ ■

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.

Wordbank ■ ■ ■ ■

■ ■ ■ ■

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 completely inside 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.

Think! Why is it impossible to construct a triangle with sides of length 7 cm, 15 cm and 5 cm?

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Start up Worksheet 9-01 Brainstarters 9

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 parallel b name two intervals that are perpendicular c 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?

Skillsheet 9-01 Line and rotational symmetry

A

B

E

D

C

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. a b c d

Worksheet 9-02 Symmetry

e

f

g

h

6 Which of the following shapes have rotational symmetry? State the order of rotational symmetry for those that do. a b c

d

e

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 MQ c NM ⊥ OP d KO II PQ. 8 a Draw a triangle that has an obtuse angle. b Draw a scalene triangle.

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f

N K

M L

R O

P

Q

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 b c d

e

h

f

i

g

j

k

l

A polygon is named by the number of sides it has. Number of sides

SkillBuilder 23-01

Pentagon

5

Shapes I

Hexagon

6

Heptagon

7

Octagon

8

Nonagon

9

Decagon

10

Undecagon

11

Dodecagon

12

Name

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.

A convex polygon

A non-convex polygon

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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.

Exercise 9-01 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? d a decagon? g a dodecagon?

b a quadrilateral? e a heptagon? h an octagon?

c a nonagon? f a pentagon? 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 d a non-regular heptagon

b a non-regular hexagon e a convex pentagon

5 Which of the following shapes are not polygons? a trapezium b ellipse d diamond e prism 6 a b c d Geometry 9-01 The vocabulary of geometry

c a regular triangle f a non-convex dodecagon. c square f circle

Draw a pentagon with one axis of symmetry. Draw a quadrilateral with four axes of symmetry Draw a hexagon with six axes of symmetry Draw a decagon with two axes of symmetry.

7 Geometry software, such as Cabri Geometry or Geometer’s 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

286

e

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f

10 Copy these composite shapes into your book and divide them into the shapes requested. a

b

c

Two pentagons Two triangles and one rectangle

d

e

Two trapeziums

f

Four triangles

One triangle and one trapezium

One trapezium and one hexagon

g

One square and one heptagon

11 Use Geometer’s Sketchpad or Cabri Geometry to create shapes that a partner can try to draw using your instructions.

Geometry 9-02 Creative copying

Working mathematically Communicating: Logos and designs 1 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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Classifying triangles A polygon is two-dimensional (flat) 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 classified in two ways: • by their sides (equilateral, isosceles or scalene) • by their angles (acute-angled, obtuse-angled, right-angled). Sides

Equilateral Three equal sides Three equal angles

Isosceles Two equal sides Two equal angles

Scalene No equal sides No equal angles

Acute-angled All three angles acute

Right-angled One right angle

Obtuse-angled One obtuse angle

Angles

Example 1 R

Classify this triangle using sides and angles.

T

S

Solution This triangle has two equal sides and one obtuse angle. It is isosceles and obtuse-angled.

Exercise 9-02 Example 1

1 Rule up a table with these headings: Acute-angled Obtuse-angled Right-angled Equilateral

Worksheet 9-03 Properties of triangles

Isosceles

Scalene

Place the letter of each of the following triangles under the headings that match. (The same triangle may appear under more than one heading.)

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a

b

c

30°

88° 62°

d

e

20°

f

140° 20°

3 cm

g

h 3 cm

i SkillBuilder 23-02

3 cm

Shapes II

k

l cm

j

12 cm 60°

15

60°

12 cm

60°

12 cm

6 cm

2 Draw the following triangles: a a scalene triangle c an isosceles triangle e a right-angled isosceles triangle

b a right-angled scalene triangle d an equilateral triangle f an acute scalene triangle.

3 Use Geometer’s Sketchpad or Cabri Geometry to explore and draw special triangles and form your own definitions.

Geometry 9-03

4 Is it possible to draw an equilateral right-angled triangle? Why?

Triangles

5 Copy these triangles into your book and draw in all axes of symmetry. a b c

d

e

f

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 prefix ‘tri’ means ‘three’. Find the meaning of each of the following mathematical ‘tri’ words: a trisect b trilateral c triangulate. GEOMETRIC FIGURES

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Working mathematically Reasoning and reflecting: Building with shapes 1 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.

Naming geometric figures Points, lines and intervals A 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.

P

A line is a straight edge that continues infinitely (forever) in both directions, so it is usually drawn with arrowheads at both ends. line A line is named by any two points on it. The line in L 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 definite length. An interval is interval labelled by its two end-points. The interval in this R diagram is RS.

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NEW CENTURY MATHS 7

A

points

O

M

S

A

Triangles A triangle is identified by the capital letters that label its vertices or angles. The triangle in this diagram is labelled
ABC or
BCA or
CAB (figures 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.

Quadrilaterals

C R

S T A b c C a The side labelled a is opposite the angle labelled A.

A quadrilateral is any plane shape with four sides and is identified 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 identified by the two capital letters labelling their endpoints. The red side can be labelled PS or SP.

Equal angles and intervals In 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.

B

B

P

Q

R S

D

F E

G

Parallel and perpendicular intervals In 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.

W

X

Z

Y

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Example 2
JKL 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’.

J

Solution a b c d

L

K

M The two smaller triangles are
JML and
JKM. JL and JK have equal length. ∠JLM and ∠JKM (or ∠L and ∠K) If side JM is perpendicular to side LK, then the lengths of intervals LM and MK are equal.

Example 3 a Draw a parallelogram and label it DEFG. c Name both pairs of parallel sides. e Mark the equal sides DG and EF.

b Mark both pairs of parallel sides. d Mark the equal angles ∠D and ∠F.

Solution a The answer should resemble the diagram on the right. D 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 G arcs on the diagram. e The equal sides DG and EF are marked by dashes on the diagram.

E

F

Exercise 9-03 Example 2

1 In this diagram, name a pair of intervals that are: a equal b perpendicular c parallel.

D

E F

G H

J

I Example 3

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.

K

M

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NEW CENTURY MATHS 7

L

4 What is the difference between the line EF and the interval EF?

F E P

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?

T Q S R C

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 b c d

D F

Draw an isosceles triangle
EFG where EF = EG. Label the sides of the triangle e, f, and g. What is another name for the side EF? Mark on the triangle the equal angles ∠F and ∠G.

E

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 b c d e

What type of quadrilateral is STUV? VS = ST. True or false? VS II ST. True or false? ˆ . True or false? SVˆ T = VTU Name the marked pair of equal sides.

S

T

V

U

10 a b c d e

Draw a square, WXYZ, and mark all equal sides and angles. Name the point where side XY meets side ZY. Name a pair of parallel sides and mark them. Name a pair of perpendicular sides. Explain the meaning of: i WX ⊥ XY ii WX = XY

11 a b c d

Draw a trapezium, UVWX, where UX = VW. Mark the equal angles UXW and VWX. UX II VW. True or false? UV II XW. True or false?

W

X

Z

Y

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Constructing triangles To 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 4 1 Construct a triangle with sides 3 cm, 5 cm and 4 cm.

4 cm

3 cm Rough sketch 5 cm

Solution Step 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.)

3c

m

5 cm

5 cm

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.)

4

cm

5 cm

Step 4: Complete the triangle by joining the intersecting point of the arcs to the ends of the interval. 3 cm

4 cm

5 cm A

2 Construct
ABC where a = 5 cm, ∠C = 30° and ∠B = 70°.

b Rough sketch C

c

30° a = 5 cm

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NEW CENTURY MATHS 7

70° B

Solution Step 1: Draw an interval 5 cm long.

5 cm

C

B

Step 2: Draw a 70° angle at B. Step 3: At C, draw a 30° angle. 70°

30° 5 cm

C

B

Step 4: Join the arms to complete the triangle.

A

70°

30° C

B

5 cm

3 Construct a triangle with one side measuring 6 cm, another side measuring 4 cm and an angle between them of 35°.

4 cm

Rough sketch 35° 6 cm

Solution Step 1: Draw an interval 6 cm long.

6 cm

Step 2: Draw an angle of 35° at one end. 35° 6 cm

Step 3: Measure an interval of 4 cm on the new arm. 4 cm 35° 6 cm

Step 4: Complete the triangle. 4 cm

35° 6 cm

Note: 35° is called the included angle because it is between the two sides.

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Exercise 9-04 1 Construct each of these triangles accurately: Q a b

4 F

40°

O

R

4 cm

d

c

C 2 cm

S

M

3 cm

cm

Example 4

e

80° 6 cm

E

N

K

3 cm

Y

f

3 cm

G

130°

mm 40

3c

m

Geometry 9-04 Constructing triangles

m 3c

3c

m

X

H

D

M

120° 3 cm

L

30°

Z

2 Geometer’s 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 is always the angle.

the largest angle, while the shortest side

5 a Which triangle in Question 1 is equilateral? b Measure its angles. What do you notice? Worksheet 9-03 Properties of triangles

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 triangle’s 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 first. 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 triangle’s angles? c Measure the angles. What do you notice? d Copy and complete: An isosceles triangle has two sides, and two angles opposite them.

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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?

Classifying quadrilaterals A 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 …

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

Diagrams

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Just for the record Velcro sticks All 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.

Example 5 1 PQRS is a parallelogram, as shown on the right. P a Measure the lengths of its sides. Are opposite sides equal? b Measure the size of its angles. Are opposite angles equal? S 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?

Q

R

Solution

a 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 P Q (you cannot fold it in half). d A parallelogram has rotational symmetry of O order 2. You can rotate it 180° so that it maps on to itself. The centre of symmetry is marked O. S R e PR = 4.2 cm while QS = 5 cm. The diagonals are not equal. A 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? D

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B

C

Solution a b c d

The diagonals, AC and BD, have equal length. The diagonals bisect each other (cut each other in half), shown by the equal markings. The diagonals do not intersect at right angles. 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.

Exercise 9-05 1 Find what the prefix ‘quad’ means. List other words beginning with ‘quad’. 2 Label each of these quadrilaterals as convex or non-convex. a b c

d

e

f

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.

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299

Worksheet 9-04 Properties of quadrilaterals

CHAPTER 9

Geometry 9-05 Quadrilaterals Example 5

4 Use Geometer’s Sketchpad or Cabri Geometry to explore the quadrilaterals in Question 3 and form your own definitions. 5 a Copy or print out this table. Trapezium

Parallelogram

Rhombus

Rectangle

Square

Kite

Opposite sides are equal

Worksheet 9-04 Properties of quadrilaterals

Opposite sides are parallel Opposite angles are equal All angles are 90° Diagonals are equal Number of axes of symmetry Order of rotational symmetry

SkillBuilder 23-05 4-sided figures

SkillBuilder 23-11 Axes of symmetry

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 parallelogram d 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.

Worksheet 9-04 Properties of quadrilaterals

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. 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

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?

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Properties of triangles and quadrilaterals: a summary Shape

Equilateral triangle

Properties

• All three sides equal • All three angles 60° • Three axes of symmetry

60° 60°

60°

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

= 45°

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Working mathematically Worksheet 9-05

Applying strategies and reasoning: Shape puzzles All of the shapes used in this activity can be printed out. Use the link to find them.

Shape puzzles

1 a How many squares can you find in this shape? (The answer is not 16!) b How many rectangles can you find?

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 five squares can be arranged. These shapes are called pentominoes. The first five 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. D

C

B A

5 How many triangles can you find in each of these shapes? a b c

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d

6 Copy this hexagon twice on to a piece of paper and then: a cut the first 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.

Constructing perpendicular and parallel lines The set square A 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: 45° 60° The 60°–30° set square The 45° set square

30° 45°

Perpendicular lines and parallel lines can be constructed using a set square or a protractor.

Example 6 Use a set square to construct a line perpendicular to XY through point W.

X

W

Y

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Solution (A perpendicular line can also be constructed using a protractor, by measuring a 90° angle.)

X

W Y

Example 7 Z

Use a set square to construct a line perpendicular to XY through point Z (where Z is not on XY).

X

Y

Solution Step 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 fits the edge of the set square and is perpendicular to XY. Step 5: Rule the perpendicular line.

Z

X

Y

Example 8 B

Use a set square to construct a line parallel to AB through point P. A

P

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NEW CENTURY MATHS 7

Solution Step 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 firmly 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.)

B

A

P

Exercise 9-06 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.

Example 6

L

4 Draw a line and mark a point X below it. Construct a perpendicular line to the line through X.

Example 7

X

5 Draw a line and mark a point P above it. Construct a parallel line through P.

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?

C

A

7 a b c d

X

Example 8

B

Draw a line XY and mark a point A above it. Draw a line parallel to XY through A. Draw a line from A to XY perpendicular to the line drawn in part b. Label the line AB. What do you notice about AB and XY? GEOMETRIC FIGURES

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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?

D G E F

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 first interval. Join the ends of both intervals to make a quadrilateral. What type of quadrilateral have you constructed?

Geometry 9-06 Parallel and perpendicular lines

11 Use the link to go to an exercise which uses Cabri Geometry or Geometer’s Sketchpad to draw parallel and perpendicular lines.

Working mathematically Communicating and reflecting: Vertical lines Many people use the word ‘perpendicular’ when they really mean vertical. A vertical line is a line that is perpendicular to the Earth’s surface (or the horizon). ‘Vertical’ means ‘up and down’, while ‘horizontal’ means ‘flat’ or ‘across’. Brick walls are vertical.

Right angle

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’.

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NEW CENTURY MATHS 7

Constructing quadrilaterals G

Worksheet 9-06 Constructions in diagrams

H

Worksheet 9-07 Constructions in words

Example 9 F

Construct a square, FGHI, of side length 4 cm. Rough sketch

I

4 cm

Solution F

Step 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.

G

Worksheet 9-08 Try drawing these!

I

H

4 cm

Example 10 P

Construct this kite, PQRS. 3.5 cm S 105°

Q

5 cm

P

5 3.

3.

cm

5

cm

S 105°

Q

5c

Draw PS of length 3.5 cm. Measure 105° at ∠S. Construct SR of length 5 cm. 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.

m

Step 1: Step 2: Step 3: Step 4:

R

m

5c

Solution

R GEOMETRIC FIGURES

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CHAPTER 9

Exercise 9-07 1 a Draw BA measuring 6 cm. b Construct a perpendicular, BC, 3 cm long. c Complete the rectangle, ABCD.

3 cm

C

B

2 Construct a square, KLMN, of side length 5 cm.

2 cm

R

S

4c

m

3 Construct this trapezium.

U

60°

T

6 cm

4 Construct a parallelogram with sides of 6 cm and 4 cm and an included angle of 65°. 4 cm

3 cm

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?

4 cm

6 a Draw two joined intervals of the same length and use your instruments to complete these shapes. 25 mm i ii 3 cm

Example 9

A

6 cm

25 mm

3 cm

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.

P

3 cm Q

4 cm 4 cm S

1 cm 5 cm

8 Construct a rhombus with sides of 6 cm and an included angle of 50°.

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NEW CENTURY MATHS 7

R

9 a Construct this quadrilateral. b What type of quadrilateral is JKLM?

5.5 cm

J

K

100° 3 cm

M

80° L

4 cm W

Example 10

10 Construct this kite. 4 cm Z

X

6 cm 40°

11 Construct the quadrilateral ABCD where AB ⊥ BC, AB = 7 cm, BC = 3 cm, DC = 5 cm and AD = 4.5 cm.

Y

12 Construct the trapezium DEFG where DE II GF, DE = 6 cm, EF = GF = 3 cm and ∠F = 135°.

Power plus 1 Name each of the following polygons and state whether it is convex or non-convex. a b c

d

e

f

2 Explain the difference between parallel lines, perpendicular lines and skew lines. 3 Use the definitions 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?

GEOMETRIC FIGURES

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CHAPTER 9

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? c a dodecagon?

b an octagon?

6 Name all the quadrilaterals whose diagonals: a bisect each other at right angles b bisect each other c intersect at right angles d have equal length e 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.

A

B

C

8 Name the most general quadrilateral in which: a opposite angles are equal b diagonals intersect at 90° c diagonals are equal d all angles are 90° e 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. X

A

B

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?

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NEW CENTURY MATHS 7

Language of maths acute-angled convex included angle line symmetry parallel protractor rhombus set square

bisect decagon interval obtuse-angled parallelogram quadrilateral right-angled square

compasses diagonal isosceles octagon perpendicular rectangle rotational symmetry trapezium

construct equilateral kite order polygon regular polygon scalene vertex/vertices

Worksheet 9-09 Geometry find-a-word

1 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 find 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?

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.

Triangles A ______ O ______ R ______ E ______ S ______ I ______

Naming geometric figures

GEOMETRIC FIGURES

Constructing figures

Polygons Quadrilaterals

GEOMETRIC FIGURES

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CHAPTER 9

Chapter 9

Review

Topic test Chapter 9

Ex 9-01

1 What type of polygon has 10 sides?

Ex 9-01

2 Name a shape that is not a polygon.

Ex 9-01

3 Draw: a a regular pentagon c a convex quadrilateral

Ex 9-02

Ex 9-02

b a non-regular pentagon d a non-convex quadrilateral.

4 Classify these triangles, by sides and angles. a b

c

d

f

e

5 a Classify
FGH by sides and angles. b Which angles in
FGH are equal?

F 5 cm

H Ex 9-03

6 a Name a pair of parallel sides in this figure. b Name a pair of perpendicular sides. c What type of quadrilateral is ABCD?

4 cm

G B

A

D Ex 9-04

4 cm

C

7 Construct the following triangles. a A 40° 6 cm 40° C

B

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. Ex 9-04

8 a Draw an obtuse-angled triangle,
XYZ, and label its sides x, y and z. b What is the relationship between the triangle’s longest side and its largest angle?

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NEW CENTURY MATHS 7

9 Name each of the following polygons: a b

d

Ex 9-05

c

e

f

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.

Ex 9-05

11 What is the definition of a rhombus?

Ex 9-05

12 Write two properties of a parallelogram.

Ex 9-05

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 five 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°.

Ex 9-05

14 Copy this diagram and use a set square and ruler to construct a line, through P: a perpendicular to QR b parallel to QR.

Ex 9-06

Q P

R

15 Construct this parallelogram.

Ex 9-07

4 cm 80° 6 cm

16 Construct this quadrilateral.

M

4 cm

3 cm Q

Ex 9-07

N 4 cm

55° 5 cm

P

GEOMETRIC FIGURES

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CHAPTER 9