Chapter 9

09 NCM7 2nd ed SB TXT.fm Page 302 Saturday, June 7, 2008 5:15 PM

SPACE AND GEOMETRY

Geometrical shapes—rectangular bricks, triangular building frames, circular wheels—surround you every day. You see them in buildings, on TV, as you drive and when you’re at school. The most common figures are triangles and quadrilaterals (four-sided shapes). This chapter focuses on those figures.

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• convex polygon A polygon • label and name points, lines, intervals, equal 1234567890123456789 5678whose vertices point 234567890123 45678901234 0123456789equal angles, parallel lines, intervals, outwards, not inwards, where any diagonal lies 0123456789 56789012 456789012345678901234567890123456789012345678 56789012 8 3456789012345678 perpendicular lines, 7890123456789012 triangles and quadrilaterals completely inside it. 789012345678 901234567890123456789012345678901234567890123456789012345678 • recognise different types of polygons, including • diagonal An interval joining two non-adjacent 567890123456789012345678901234567890123456789012345 0123456789 56789012345678901234567890 1234567890123456789 convex and regular polygons vertices of a polygon. 78901234 5678 234567890123 45678901234 0123456789 • recognise and classify triangles using sides and • included angle The angle between two given 45678901 6789012345678901234567890123456 7890123456789012 345678901 angles sides of a polygon. 456789012 9012345 67890123456789012345678901234567890123456789012345678901234567890123 6789012345678901234567890123456789012345678901234567890123 • recognise and classify quadrilaterals, including • interval Part of a line, with a starting point, an end 7890123456789012 89012345678 345678 9012345 67890123456789012345678 90123456789012345678901234567890123456 345678901234567890123 901234567890123 01234567890 123456789012345678901234 012345678901234 5678901 convex and non-convex quadrilaterals point and a definite length. 5678 0123 4567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678 45678901234567890123456 789012345678901234567 8901234 678901234567890 2345678901 234567 • construct perpendicular lines and parallel lines • obtuse-angled triangle A triangle with one 756789012345678901234 45678901234 6789012345678901 34567890123456789012345678901234567 0123456789012345 using set squares and rulers obtuse 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09 NCM7 2nd ed SB TXT.fm Page 304 Saturday, June 7, 2008 5:15 PM

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 B a name two intervals that are parallel b name two intervals that are perpendicular E c are the diagonals AC and DB equal in length? D C 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°?

Skillsheet 9-01

5 Copy these shapes and mark in the axes of symmetry on each one. a b c d

Line and rotational symmetry

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 O

R

M L P

Q

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9-01 Naming geometrical figures

Geometry 9-01 The vocabulary of geometry

Points, lines and intervals A point is a position represented by a dot which is labelled by a capital letter.

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 this diagram is labelled LM.

M

L

An interval is a part of a line. It has a starting point, an end point and a definite length. The interval in this diagram is RS.

interval R

S

Triangles A

A triangle is identified by the capital letters that label its angles. The triangle in this diagram can be labelled ABC or BCA or CAB.

C 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.

B

A

This diagram shows angles 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 purple side can be called CB, BC or a.

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

a

B

Quadrilaterals A quadrilateral is any plane shape with four sides and is identified by the capital letters that label its angles. The quadrilateral in this diagram is labelled PQRS or QRSP or RSPQ or SPQR. The sides of a quadrilateral can be identified by the two capital letters labelling their endpoints. The purple side can be labelled PS or SP.

Equal angles and intervals

P

Q

R S D

In geometrical diagrams, equal angles are marked by identical symbols, while equal intervals are marked by identical dashes. In this diagram, DF, DE and EG are all the same length while ∠F and ∠DEF are the same size.

F E

CHAPTER 9 GEOMETRICAL FIGURES

G

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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

Example 1 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

L

K

M

Solution a 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.

Example 2 a b c d e

Draw a parallelogram and label it DEFG. Mark both pairs of parallel sides. Name both pairs of parallel sides. Mark the equal angles ∠D and ∠F. Mark the equal sides DG and EF.

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. G d The equal angles ∠D and ∠F are marked by equal arcs on the diagram. e The equal sides are marked on the diagram with two dashes.

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E

F

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Exercise 9-01 1 In this diagram, name a pair of intervals that are: D a equal b perpendicular G c parallel.

E

Ex 1

F Geometry 9-02 Copy my sketch

H

J

I

2 a Draw two lines, PQ and RS, intersecting at T. b Mark the equal angles ∠PTR and ∠STQ. c ∠PTR = ∠STQ. Why?

Ex 2

3 Copy the triangle KLM shown on the right, and correctly label its sides k, l, and m.

K

M

L

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 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.

F E P T Q S R C D F

7 a Draw an isosceles triangle EFG where EF = EG.

E

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?

CHAPTER 9 GEOMETRICAL FIGURES

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9 a VS = ST. True or false? c VS II ST. True or false? ˆ . True or false? ˆ T = VTU d SV

T U S

e Name the marked pair of equal intervals.

V

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

W

X

Z

Y

9-02 Classifying polygons A polygon is a 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. Name

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NEW CENTURY MATHS 7

Number of sides

Pentagon

5

Hexagon

6

Heptagon

7

Octagon

8

Nonagon

9

Decagon

10

Undecagon

11

Dodecagon

12

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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 all point outwards while non-convex (or concave) polygons have some vertices that point or cave inwards.

A convex polygon

A non-convex polygon

A simple test to determine if a polygon is convex or non-convex is to draw any diagonal joining two vertices of the polygon. If all diagonals lie completely inside If all or part of any diagonal lies outside the polygon, then it is convex. of the polygon, then it is non-convex.

Exercise 9-02 1 Name each of the polygons (a to l) on page 308, 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 Which shape is a regular convex octagon? Select A, B, C or D. A B C

D

5 Draw the following shapes. a a regular hexagon b a non-regular hexagon d a non-regular heptagon e a convex pentagon

c a regular triangle f a non-convex dodecagon

6 Which of the following shapes are not polygons? a a trapezium b an ellipse d a diamond e a prism

c a square f a circle

7 a b c d

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. CHAPTER 9 GEOMETRICAL FIGURES

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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

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

Two pentagons

d

Two triangles and one rectangle

e

Two trapeziums

f

Four triangles

Working mathematically

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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NEW CENTURY MATHS 7

One triangle and one trapezium

Communicating

One trapezium and one hexagon

g

One square and one heptagon

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9-03 Classifying triangles 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).

Worksheet 9-03 Properties of triangles

Sides

Equilateral Three equal sides

Isosceles Two equal sides

Scalene No equal sides

Acute-angled All three angles acute

Right-angled One right angle

Obtuse-angled One obtuse angle

Angles

Example 3 Classify this triangle using sides and angles.

R

T

S

Solution The triangle has two equal sides so it is isosceles. The triangle has an obtuse angle so it is obtuse angled. So RST is isosceles and obtuse-angled.

Exercise 9-03 1 Rule up a table with these headings. Acute-angled

Obtuse-angled

Right-angled

Ex 3

Equilateral

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.)

CHAPTER 9 GEOMETRICAL FIGURES

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a

b

c

d

30°

88° 62°

e

f 140°

3 cm

g

h

20° 3 cm

3 cm

20°

j

l 12 c m

cm

60°

12

60°

cm

k

15

i

12 cm

60° 6 cm

2 Draw the following triangles. a a scalene triangle c an isosceles triangle e a right-angled isosceles triangle Geometry 9-03 Triangles

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

3 Use The Geometer’s Sketchpad or Cabri Geometry to explore special triangles. 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. 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 Which triangle is an obtuse-angled scalene triangle? Select A, B, C or D. A

B

C

D

9 The prefix ‘tri’ means ‘three’. Find the meaning of these mathematical ‘tri’ words. a trisect b trilateral c triangulate

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Mental skills 9

Maths without calculators

Simplifying multiplication by factorising Sometimes, one big multiplication may be made into many simpler little multiplications if we break each term into two ‘easier’ factors. Then we use the property that numbers can be multiplied in any order. 1 Examine these examples. a 14 × 36 = 7 × 2 × 6 × 6 b 15 × 6 = 3 × 5 × 3 × 2 =6×7×6×2 =3×3×5×2 = 42 × 6 × 2 = 9 × 10 = 252 × 2 = 90 = 504 c 30 × 22 = 3 × 10 × 2 × 11 d 24 × 8 = 6 × 4 × 2 × 4 = 3 × 2 × 10 × 11 =4×4×6×2 = 6 × 11 × 10 = 16 × 6 × 2 = 660 = 96 × 2 = 192 Look for different pairs of numbers you can multiply easily and rearrange. Note: For each question, there are many different possible ways of arriving at the correct result. 2 Now find these products by factorising first. a 30 × 24 b 18 × 27 c 25 × 33 e 28 × 20 f 12 × 18 g 21 × 9 i 16 × 35 j 22 × 28 k 9 × 15

d 16 × 8 h 36 × 12 l 27 × 25

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.

CHAPTER 9 GEOMETRICAL 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 on the right. 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.

9-04 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

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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.)

4c

m

5 cm

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

4c

3c

m

5 cm

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

A Rough sketch C

b c

30° a = 5 cm

Solution Step 1: Draw an interval 5 cm long.

70° B

5 cm

C

B

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

70° 5 cm

C

B A

Step 4: Join the arms to complete the triangle.

30° C

70° 5 cm

CHAPTER 9 GEOMETRICAL FIGURES

B

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

4c

m

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.

4c

m 35°

6 cm

Step 4: Complete the triangle. 4c

m 35°

6 cm

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

Exercise 9-04

O

4

R

4 cm F

d

c

C 2 cm

S

M

3 cm

cm

1 Construct each of these triangles accurately. Q a b

e

40°

80°

N

6 cm

K

E

3 cm

40 mm

m

m

3c

3c

m

H

316

3 cm

G

NEW CENTURY MATHS 7

120° M

3 cm

L

Z

D Y

f X

3c

Ex 4

130°

30°

09 NCM7 2nd ed SB TXT.fm Page 317 Saturday, June 7, 2008 5:15 PM

2 The 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.

Geometry 9-04 Constructing triangles

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 shortest side is always the

the largest angle, while 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.

Worksheet 9-03

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. 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?

CHAPTER 9 GEOMETRICAL FIGURES

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Properties of triangles

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11 Copy this table of properties of a triangle into your book and complete it. Summary: Properties of triangles Shape Equilateral triangle

Properties • All three sides are 60°

• All three angles measure • Has

60°

axes of symmetry

60°

Isosceles triangle

• Two sides are • Two

equal (opposite the equal sides)

• Has one axis of

Scalene triangle

Worksheet 9-04 Properties of quadrilaterals

• No • Has

or angles are equal axes of symmetry

9-05 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

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NEW CENTURY MATHS 7

Diagrams

09 NCM7 2nd ed SB TXT.fm Page 319 Saturday, June 7, 2008 5:15 PM

Square

four equal sides and four right angles

Kite

two pairs of adjacent sides equal

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 S 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?

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. 2 This diagram illustrates the properties of the A 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

B

C

Solution a 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.

CHAPTER 9 GEOMETRICAL FIGURES

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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 Name each quadrilateral in this diagram. c b

a

e

d

Kite

Square

Rectangle

5 a Cut out an example of each quadrilateral in Question 3. Copy this table or print out Worksheet 9-04.

Rhombus

Ex 5

Parallelogram

Quadrilaterals

f

4 Follow the link to use The Geometer’s Sketchpad or Cabri Geometry to explore the quadrilaterals in Question 3.

Trapezium

Geometry 9-05

Opposite sides are equal

Worksheet 9-04

Opposite sides are parallel

Properties of quadrilaterals

Opposite angles are equal All angles are 90° Diagonals are equal Number of axes of symmetry Order of rotational 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).

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NEW CENTURY MATHS 7

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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

Parallelogram

Properties of quadrilaterals

Kite

Square

Rectangle

Worksheet 9-04

Rhombus

8 Use a ruler and protractor with the quadrilaterals you cut out in Question 5, 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

7 List the quadrilaterals in Question 6 that have rotational symmetry and state the order of rotational symmetry for each.

Diagonals are equal Diagonals bisect each other Diagonals intersect at right angles Diagonals bisect angles of quadrilateral

9 I am a quadrilateral with opposite sides equal and parallel. My diagonals are equal and I have four axes of symmetry. Which quadrilateral am I? Select A, B, C or D. A rectangle B square C parallelogram D rhombus 10 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. 11 a Does a square have all the properties of a rectangle? Why? b Does a rhombus have all the properties of a parallelogram? Why? 12 Copy this table of properties of quadrilaterals into your book and complete it. Summary: Properties of quadrilaterals Shape

Properties

Trapezium

• One pair of • No axes of symmetry

Kite

•

sides

pairs of adjacent sides are equal

• One pair of opposite angles are • Has

axis of symmetry

• Diagonals intersect at Parallelogram

•

angles

sides are equal and parallel

• Opposite angles are • No axes of symmetry • Diagonals

each other

CHAPTER 9 GEOMETRICAL FIGURES

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Shape

Properties

Rhombus

• All four sides are • sides are parallel • Opposite angles are • Has axes of symmetry • bisect at right angles • Diagonals bisect the of the rhombus

Rectangle

• All four angles measure • Opposite sides are and • Has two axes of • Diagonals are • bisect each other

Square

• All four sides are • All four angles measure • Has axes of symmetry • Diagonals are equal and each other at right angles • bisect the angles of the square

Working mathematically

Applying strategies and reasoning

Shape puzzles Worksheet 9-05 Shape puzzles

All of the shapes used in this activity can be printed out on Worksheet 9-05. 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 to join five squares edge to edge. These shapes are called pentominoes. Here are five of them. Draw the other seven.

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NEW CENTURY MATHS 7

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4 Copy this equilateral triangle, cut out the four pieces and rearrange them into a square. D

B

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

C

A

d

6 Copy the 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.

9-06 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. CHAPTER 9 GEOMETRICAL FIGURES

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

X

W

Solution

Y

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

X

W

Y

Example 7 Use a set square to construct a line perpendicular to XY through point Z (where Z is not on XY).

Z

X

Solution Step 1: Place the ruler through the point Z. Step 2: Slide the set square along XY until it meets the ruler. Step 3: Slide the ruler until it fits the edge of the set square and is perpendicular to XY. Step 4: Remove the set square and rule the perpendicular line.

Y Z

X

Y

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NEW CENTURY MATHS 7

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Example 8 Use a set square to construct a line parallel to AB through point P.

B

A

P

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. The construction works because corresponding angles on parallel lines are equal.

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.

L

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

X

Ex 6

Ex 7

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?

P

Ex 8

C

A

X

CHAPTER 9 GEOMETRICAL FIGURES

B

325

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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. 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? 11 What is the difference in meaning between ‘perpendicular’ and ‘vertical’? Geometry 9-06

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

Parallel and perpendicular lines

9-07 Constructing quadrilaterals Example 9 Worksheet 9-06

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

F

G

Constructions in diagrams

Rough sketch Worksheet 9-07

I

Constructions in words

Worksheet 9-08 Try drawing these!

Solution 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.

F

I

326

NEW CENTURY MATHS 7

4 cm

H G

4 cm

H

09 NCM7 2nd ed SB TXT.fm Page 327 Saturday, June 7, 2008 5:15 PM

Example 10 Construct this kite, PQRS.

P 3.5 cm S 105°

Q

5 cm R

P

5 3.

3.

cm

5

cm

Q

m

S 105°

m

5c

5c

Solution Step 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 compasses to draw an arc of radius 5 cm. Step 5: At P, use your compasses 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.

R

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

A

6 cm

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

Ex 9

3 Construct the trapezium shown on the right.

2 cm

S

4 cm

R

U

60° 6 cm

CHAPTER 9 GEOMETRICAL FIGURES

T

327

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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?

4 cm

3 cm 4 cm

3 cm

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

25 mm

3 cm

b What is each shape 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.

3 cm

P

Q

4 cm 4 cm 1 cm

S

R

5 cm

8 Construct a rhombus with sides of 6 cm and an included angle of 50°. 9 a Construct this quadrilateral.

J

b What type of quadrilateral is JKLM?

5.5 cm

K

100°

3 cm

M Ex 10

80° 4 cm

10 Construct the kite shown on the right.

L

W 4 cm X

Z

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°.

328

NEW CENTURY MATHS 7

6 cm 40° Y

09 NCM7 2nd ed SB TXT.fm Page 329 Saturday, June 7, 2008 5:15 PM

Using technology

More spreadsheet formulas involving integers Set up a spreadsheet as shown below, where m represents the value in cell B1, n is the value in cell B2, p is the value in cell B3, and so on.

Enter the following formulas into the given cells. (Try to predict each answer before you enter the formula.) a C1, m + n (means enter the formula into cell C1) b C2, m − n c C3, p + r d C4, n × p e C5, p ÷ r f C6, m − p − q g C7, q ÷ n × r h C8, m × n + q ÷ r i C9, the sum of all cells B1 to B5 j C10, the maximum value from the set of cells B1 to B5 j C12, m2 + p2 + q2 k C11, 10 × (p × r) ÷ (m + n)

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 pages 318-19 to help you answer these questions. a Is the square a special type of rhombus? b Is the rhombus a special type of square?

CHAPTER 9 GEOMETRICAL FIGURES

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c d e f

Is the parallelogram a special type of trapezium? Is the rectangle a special type of parallelogram? Is the parallelogram a special type of kite? Is the rectangle a special type of square?

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 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 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.

C

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

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Chapter 9 review Language of maths acute-angled convex included angle line symmetry parallel protractor rhombus set square

Worksheet 9-09

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

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 geometrical figures

GEOMETRICAL FIGURES

Constructing figures

Polygons Quadrilaterals

CHAPTER 9 GEOMETRICAL FIGURES

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Topic test 9

Chapter revision 1 a Name a pair of parallel sides in this figure. b Name a pair of perpendicular sides. c Name a pair of equal angles.

A

Exercise 9-02

2 What type of polygon has 10 sides?

D

Exercise 9-02

3 Name a shape that is not a polygon.

Exercise 9-02

4 Draw the following shapes. a a regular pentagon c a convex quadrilateral

Exercise 9-01

Exercise 9-03

c

e

f

6 a Classify FGH by sides and angles.

F

b Which angles in FGH are equal?

5 cm

H Exercise 9-04

C

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

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

d

Exercise 9-03

B

4 cm

4 cm

G

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. Exercise 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?

332

NEW CENTURY MATHS 7

09 NCM7 2nd ed SB TXT.fm Page 333 Saturday, June 7, 2008 5:15 PM

9 Name each of the following quadrilaterals. a b

d

Exercise 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.

Exercise 9-05

11 What is the definition of a rhombus?

Exercise 9-05

12 Write two properties of a parallelogram.

Exercise 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°.

Exercise 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.

Q

Exercise 9-06

P

R

15 Construct this parallelogram.

Exercise 9-07

4 cm 80° 6 cm

16 Construct this quadrilateral.

M

4 cm

Exercise 9-07

N 4 cm

3 cm 55° Q

5 cm

P

CHAPTER 9 GEOMETRICAL FIGURES

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Mixed revision 3 Exercise 7-03

Exercise 7-05

Exercise 7-08

1 Draw a number line labelled from 0 to 3, and mark these decimals on it. 0.5, 2.25, 1.3, 2.9, 0.75, 1.6 2 Find: a 1.54 + 7.9 d 6.31 + 0.2 + 15.38

b 9.2 − 6.1 e 0.003 + 1.01 + 4.394

c 22.6 − 13.55 f 9.803 − 4.52

3 Find: a 6.2 × 0.7 d 4.21 × 0.5

b 60.138 ÷ 6 e 0.05 × 0.4

c 1.672 ÷ 0.02 f 4.032 ÷ 0.3

Exercise 7-09

4 Sandra bought 40 litres of LPG at 79.9c per litre. Catherine bought 40 litres at 87.9c per litre. How much more than Sandra did Catherine pay?

Exercise 7-12

5 Write each of these as a decimal. a 1--b 3--5

Exercise 7-13

c 5---

8

6 Round each of these as indicated. a 0.753 to one decimal place c $9.82 to the nearest 5 cents

9

b 12.064 to two decimal places d 3.795 to two decimal places

Exercise 7-14

7 If the temperature at Penrith was 20.6ºC at 9:00am and was 29.3ºC at 3:00pm, by how much did it rise?

Exercise 7-14

8 A piece of cloth 2.5 m long was cut into 10 equal pieces. How long was each piece?

Exercise 8-01

9 What unit would you use to measure: a the distance from Sydney to Broken Hill? c the area of a football field? e the area of a page of this textbook?

Exercise 8-02

Exercise 8-04

Exercise 8-06

10 How many: a cm in 2 m? c cm in 375 mm?

b the width of a shelf? d the perimeter of a sporting field?

b mm in 12 cm? d km in 1250 m?

11 a What is the mass shown on the scale on the right? b What is the size of one unit on this scale? c What are the limits of accuracy of this scale?

75

80

85

90

kg

12 Find the perimeter of each of the following. a b 4 mm

40 cm

8 mm

c

12 cm

13 cm

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NEW CENTURY MATHS 7

9 cm

d 5 cm

9 cm

10 cm 4 cm

95

09 NCM7 2nd ed SB TXT.fm Page 335 Saturday, June 7, 2008 5:15 PM

13 A sheet of A4 paper has a width of 21.1 cm and a length of 29.7 cm. Find its perimeter. 14 How many: a cm2 in 3 m2? c m2 in 62 000 cm2?

Exercise 8-06 Exercise 8-08

mm2

cm2?

b in 9.5 d cm2 in 740 mm2?

15 Find the area of each of the shapes in Question 12. 16 A square has an area of 64

cm2.

Exercise 8-09

How long is each side? Find its perimeter.

Exercise 8-09

17 A rectangle has an area of 20 cm2. If its length is 5 cm, how wide is it? 18 Name the following shapes. a b

Exercise 8-09 Exercise 9-02

c

d

e

f

g

h

i

j

k

l

19 What is my name? a I am a three-sided figure with two of my sides equal in length. b I am a four-sided figure with all my sides equal in length. c I have four sides. My opposite sides are parallel and I have one right angle. d I am a three-sided figure with none of my angles equal. e I have ten sides.

Exercise 9-03 Exercise 9-05

20 a Construct an isosceles triangle ABC, with AB = AC = 10 cm and BC = 5 cm. Mark the two equal angles. b Draw an interval of 6 cm. Label it AB. Draw an angle of 55º at A. Draw an angle of 75º at B. Complete the triangle and label the third vertex C. Use BC to construct the triangle BCD with BD = 7 cm and CD = 5 cm. c Construct this parallelogram, PQRS, and P Q check whether its diagonals bisect each other. 4 cm 65° S

7 cm

MIXED REVISION 3

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335

Exercise 9-04

Exercise 9-07