Maths in Focus - Margaret Grove - ch4

TERMINOLOGY

4 Geometry 1

Altitude: Height. Any line segment from a vertex to the opposite side of a polygon that is perpendicular to that side

Congruent triangles: Identical triangles that are the same shape and size. Corresponding sides and angles are equal. The symbol is /

Interval: Part of a line including the endpoints

Median: A line segment that joins a vertex to the opposite side of a triangle that bisects that side

Perpendicular: A line that is at right angles to another line. The symbol is =

Polygon: General term for a many sided plane fi gure. A closed plane (two dimensional) fi gure with straight sides

Quadrilateral: A four-sided closed fi gure such as a square, rectangle, trapezium etc.

Similar triangles: Triangles that are the same shape but different sizes. The symbol is yz

Vertex: The point where three planes meet. The corner of a fi gure

Vertically opposite angles: Angles that are formed opposite each other when two lines intersect

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137Chapter 4 Geometry 1

INTRODUCTION

GEOMETRY IS USED IN many areas, including surveying, building and graphics. These fi elds all require a knowledge of angles, parallel lines and so on, and how to measure them. In this chapter, you will study angles, parallel lines, triangles, types of quadrilaterals and general polygons.

Many exercises in this chapter on geometry need you to prove something or give reasons for your answers. The solutions to geometry proofs only give one method , but other methods are also acceptable .

DID YOU KNOW?

Geometry means measurement of the earth and comes from Greek. Geometry was used in ancient civilisations such as Babylonia. However, it was the Greeks who formalised the study of geometry, in the period between 500 BC and AD 300.

Notation

In order to show reasons for exercises, you must know how to name fi gures correctly.

• B The point is called B .

The interval (part of a line) is called AB or BA .

If AB and CD are parallel lines, we write .AB CD<

This angle is named BAC+ or .CAB+ It can sometimes be named .A+

Angles can also be written as BAC^ or BAC

This triangle is named .ABC3

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138 Maths In Focus Mathematics Preliminary Course

This quadrilateral is called ABCD .

Line AB is produced to C .

DB bisects .ABC+

AM is a median of .ABCD

AP is an altitude of .ABCD

Types of Angles

Acute angle

0 90xc c c1 1

To name a quadrilateral, go around it: for example, BCDA is correct, but ACBD is not.

Producing a line is the same as extending it.

ABD+ and DBC+ are equal.

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Right angle

A right angle is .90c Complementary angles are angles whose sum is .90c

Obtuse angle

x90 180c c c1 1

Straight angle

A straight angle is .180c Supplementary angles are angles whose sum is .180c

Refl ex angle

x180 360c c c1 1

Angle of revolution

An angle of revolution is .360c

Vertically opposite angles

AEC+ and DEB+ are called vertically opposite angles . AED+ and CEB+ are also vertically opposite angles.

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Proof

( )

( ) ( )

( )

AEC x

AED x CED

DEB x AEB

x

CEB x CED

AEC DEB AED CEB

180 180

180 180 180

180 180

Let

Then straight angle,

Now straight angle,

Also straight angle,

and `

c

c c c

c c c c

c

c c c

+

+ +

+ +

+ +

+ + + +

=

= -

= - -

=

= -

= =

EXAMPLES

Find the values of all pronumerals, giving reasons.

1.

Solution

( )x ABC

x

x

154 180 180

154 180

26

154 154

is a straight angle,

`

c++ =

+ =

=

- -

2.

Solution

( )x

x

x

x

x

x

2 142 90 360 360

2 232 360

2 232 360

2 128

2 128

64

232 232

2 2

angle of revolution, c+ + =

+ =

+ =

=

=

=

- -

Vertically opposite angles are equal.

That is, AEC DEB+ += and .AED CEB+ +=

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3.

Solution

( )y y

y

y

y

y

y

2 30 90 90

3 30 90

3 30 90

3 60

3 60

20

30 30

3 3

right angle, c+ + =

+ =

+ =

=

=

=

- -

4.

Solution

(

( )

(

x WZX YZV

x

x

y XZY

w WZY XZV

50 165

50 165

115

180 165 180

15

15

50 50

and vertically opposite)

straight angle,

and vertically opposite)

c

+ +

+

+ +

+ =

+ =

=

= -

=

=

- -

5.

CONTINUED

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Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143

vertically opposite angles

straight angle,


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -

6. Find the supplement of .57 12c l

Solution

Supplementary angles add up to .180c So the supplement of 57 12c l is .180 57 12 1 2 482c c c- =l l

7. Prove that AB and CD are straight lines.

Solution

x x x xx

x

x

x

6 10 30 5 30 2 10 36014 80 360

14 280

14 280

20

80 80

14 14

angle of revolution+ + + + + + + =

+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

#

c

c

c

c

+

+

= +

=

= +

=

These are equal vertically opposite angles . AB and CD are straight lines

C

DA

B

E(2x22 +10)c

(6x+10)c

(5x+30)c

(x+30)c

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4.1 Exercises

1. Find values of all pronumerals, giving reasons.

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2. Find the supplement of (a) 59c (b) 107 31c l (c) 45 12c l

3. Find the complement of (a) 48c (b) 34 23c l (c) 16 57c l

4. Find the (i) complement and (ii) supplement of

(a) 43c 81c(b) 27c(c) (d) 55c (e) 38c (f) 74 53c l (g) 42 24c l (h) 17 39c l (i) 63 49c l (j) 51 9c l

5. (a) Evaluate x . Find the complement of (b) x . Find the supplement of (c) x.

(2x+30)c

142c

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6. Find the values of all pronumerals, giving reasons for each step of your working.

(a)

(b)

(c)

(d)

(e)

(f)

7.

Prove that AC and DE are straight lines.

8.

Prove that CD bisects .AFE+

9. Prove that AC is a straight line.

A

B

C

D

(110-3x)c

(3x+70)c

10. Show that + AED is a right angle.

A B

C

DE

(50-8y)c

(5y-20)c

(3y+60)c

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Parallel Lines

When a transversal cuts two lines, it forms pairs of angles. When the two lines are parallel, these pairs of angles have special properties.

Alternate angles

Alternate angles form a Z shape. Can you fi nd another set of

alternate angles?

Corresponding angles form an F shape. There are 4 pairs

of corresponding angles. Can you fi nd them?

If the lines are parallel, then alternate angles are equal.

Corresponding angles

If the lines are parallel, then corresponding angles are equal.

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Cointerior angles

Cointerior angles form a U shape. Can you fi nd another pair?

If AEF EFD,+ += then AB CD.<

If BEF DFG,+ += then AB CD.<

If BEF DFE 180 ,c+ ++ = then AB CD.<

If the lines are parallel, cointerior angles are supplementary (i.e. their sum is 180c ).

Tests for parallel lines

If alternate angles are equal, then the lines are parallel.

If corresponding angles are equal, then the lines are parallel.

If cointerior angles are supplementary, then the lines are parallel.

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EXAMPLES

1. Find the value of y , giving reasons for each step of your working.

Solution

( )

55 ( , )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle

corresponding angles,`

c c

c

c

+ +

+ + <

= -

=

=

2. Prove .EF GH<

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =

But CBF+ and HCD+ are corresponding angles EF GH` <

Can you prove this in a different way?

If 2 lines are both parallel to a third line, then the 3 lines are parallel to each other. That is, if AB CD< and ,EF CD< then .AB EF<

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1. Find values of all pronumerals. (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2. Prove .AB CD< (a)

(b)

A

B C

D

E104c76c

(c)

4.2 Exercises Think about the reasons for each step of your calculations.

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Types of Triangles

Names of triangles

A scalene triangle has no two sides or angles equal.

A right (or right-angled) triangle contains a right angle.

The side opposite the right angle (the longest side) is called the hypotenuse.

An isosceles triangle has two equal sides.

A

B

C

D

E

F

52c

128c

(d) AB

C

DE F

G

H

138c

115c23c

(e)

The angles (called the base angles) opposite the equal sides in an isosceles triangle are equal.

An equilateral triangle has three equal sides and angles.

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All the angles are acute in an acute-angled triangle.

An obtuse-angled triangle contains an obtuse angle.

Angle sum of a triangle

The sum of the interior angles in any triangle is 180c ,that is, a b c 180+ + =

Proof

, YXZ a XYZ b YZX cLet andc c c+ + += = =

( , , )( )

( )

AB YZ

BXZ c BXZ XZY AB YZAXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line

Then alternate anglessimilarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

<

<=

=

+ + =

+ + =

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Exterior angle of a triangle

Class Investigation

Could you prove the base angles in an isosceles triangle are equal? 1. Can there be more than one obtuse angle in a triangle? 2. Could you prove that each angle in an equilateral triangle is 3. ?60c Can a right-angled triangle be an obtuse-angled triangle? 4. Can you fi nd an isosceles triangle with a right angle in it? 5.

The exterior angle in any triangle is equal to the sum of the two opposite interior angles. That is,

x y z+ =

Proof

,

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

<

= = =

( , , )

( , , )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CEz x y

corresponding angles

alternate angles`

c

c

c

+ +

+ + +

+ + +

<

<

= +

=

=

= +

EXAMPLES

Find the values of all pronumerals, giving reasons for each step. 1.

CONTINUED

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Solution

( )x

x

x

x

53 82 180 180135 180

135 180

45

135 135

angle sum of cD+ + =

+ =

+ =

=

- -

2.

Solution

( )A C x base angles of isosceles+ + D= =

( )x x

x

x

x

x

x

48 180 1802 48 180

2 48 180

2 132

2 132

66

48 48

2 2

angle sum in a cD+ + =

+ =

+ =

=

=

=

- -

3.

Solution

)y

y

y

35 14135 141

106

35 35(exterior angle of

`

D+ =

+ =

=

- -

This example can be done using the interior sum of angles.

( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 18074 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -

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1. Find the values of all pronumerals.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

2. Show that each angle in an equilateral triangle is .60c

3. Find ACB+ in terms of x .

4.3 Exercises Think of the reasons for each step of your

calculations.

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4. Prove .AB ED<

5. Show ABCD is isosceles.

6. Line CE bisects .BCD+ Find the value of y , giving reasons.

7. Evaluate all pronumerals, giving reasons for your working. (a)

(b)

(c)

(d)

8. Prove IJLD is equilateral and JKLD is isosceles.

9. In triangle BCD below, .BC BD= Prove AB ED .

A

B

C

D

E

88c

46c

10. Prove that .MN QP

P

N

M

O

Q

32c

75c

73c

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Congruent Triangles

Two triangles are congruent if they are the same shape and size. All pairs of corresponding sides and angles are equal.

For example:

We write .ABC XYZ/D D

Tests

To prove that two triangles are congruent, we only need to prove that certain combinations of sides or angles are equal.

Two triangles are congruent if • SSS : all three pairs of corresponding sides are equal • SAS : two pairs of corresponding sides and their included angles are

equal • AAS : two pairs of angles and one pair of corresponding sides are equal • RHS : both have a right angle, their hypotenuses are equal and one

other pair of corresponding sides are equal

EXAMPLES

1. Prove that OTS OQP/D D where O is the centre of the circle.

CONTINUED

The included angle is the angle between the 2 sides.

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Solution

:

:

:

,

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)

(vertically opposite angles)

(equal radii)

by SAS`

+ +

/D D

=

=

=

2. Which two triangles are congruent?

Solution

To fi nd corresponding sides, look at each side in relation to the angles. For example, one set of corresponding sides is AB , DF , GH and JL . ABC JKL A(by S S)/D D

3. Show that triangles ABC and DEC are congruent. Hence prove that .AB ED=

Solution

: ( ): ( )

: ( )

( )

AA

S

BAC CDE AB EDABC CED

AC CD

ABC DEC

AB ED

alternate angles,similarly

given

by AAS,

corresponding sides in congruent s

`

`

+ +

+ +

<

/D D

D

=

=

=

=

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1. Are these triangles congruent? If they are, prove that they are congruent. (a)

(b)

X

Z

Y

B

C

A

4.7 m2.3 m

2.3 m

4.7 m110c 110c

(c)

(d)

(e)

(e)

2. Prove that these triangles are congruent. (a)

(b)

(c)

(d)

(e)

4.4 Exercises

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3. Prove that (a) Δ ABD is congruent to Δ ACD

(b) AB bisects BC , given ABCD is isosceles with .AB AC=

4. Prove that triangles ABD and CDB are congruent. Hence prove that .AD BC=

5. In the circle below, O is the centre of the circle.

O

A

B

D

C

Prove that (a) OABT and OCDT are congruent.

Show that (b) .AB CD=

6. In the kite ABCD, AB AD= and .BC DC=

A

B D

C

Prove that (a) ABCT and ADCT are congruent.

Show that (b) .ABC ADC+ +=

7. The centre of a circle is O and AC is perpendicular to OB .

O

A

B

C

Show that (a) OABT and OBCT are congruent.

Prove that (b) .ABC 90c+ =

8. ABCF is a trapezium with AF BC= and .FE CD= AE and BD are perpendicular to FC.

D

A B

CFE

Show that (a) AFET and BCDT are congruent.

Prove that (b) .AFE BCD+ +=

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9. The circle below has centre O and OB bisects chord AC .

O

A

B

C

Prove that (a) OABT is congruent to .OBCT

Prove that (b) OB is perpendicular to AC.

10. ABCD is a rectangle as shown below.

D

A B

C

Prove that (a) ADCT is congruent to BCDT .

Show that diagonals (b) AC and BD are equal .

Investigation

The triangle is used in many structures, for example trestle tables, stepladders and roofs.

Find out how many different ways the triangle is used in the building industry. Visit a building site, or interview a carpenter. Write a report on what you fi nd.

Similar Triangles

Triangles, for example ABC and XYZ , are similar if they are the same shape but different sizes .

As in the example, all three pairs of corresponding angles are equal. All three pairs of corresponding sides are in proportion (in the same ratio).

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Application

Similar fi gures are used in many areas, including maps, scale drawings, models and enlargements.

EXAMPLE

1. Find the values of x and y in similar triangles CBA and XYZ .

Solution

First check which sides correspond to one another (by looking at their relationships to the angles). YZ and BA , XZ and CA , and XY and CB are corresponding sides.

. .

.

. . .

CAXZ

CBXY

y

y4 9 3 6

5 4

3 6 4 9 5 4

`

#

=

=

=

We write: XYZ; DABC <D XYZD is three times larger than .ABCD

ABXY

ACXZ

BCYZ

ABXY

ACXZ

BCYZ

26 3

412 3

515 3

`

= =

= =

= =

= =

This shows that all 3 pairs of sides are in proportion.

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.. .

.

. ..

. . .

.. .

.

y

BAYZ

CBXY

x

x

x

3 64 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 62 3 5 4

3 45

#

#

#

=

=

=

=

=

=

=

Two triangles are similar if: three pairs of • corresponding angles are equal three pairs of • corresponding sides are in proportion two pairs of • sides are in proportion and their included angles are equal

If 2 pairs of angles are equal then the third

pair must also be equal.

EXAMPLES

1. Prove that triangles (a) ABC and ADE are similar. Hence fi nd the value of (b) y , to 1 decimal place.

Solution

(a) A+ is common

ADE; D

( )( )( )

ABC ADE BC DEACB AED

ABC

corresponding angles,similarly3 pairs of angles equal`

+ ++ +

<

<D

=

=

(b)

CONTINUED

Tests

There are three tests for similar triangles.

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. .

.

. .. . .

.. .

.

.

AE

BCDE

ACAE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

#

#

= +

=

=

=

=

=

=

2. Prove .WVZD;XYZ <D

Solution

( )

ZVXZ

ZWYZ

ZVXZ

ZWYZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=

since two pairs of sides are in proportion and their included angles are equal the triangles are similar

Ratio of intercepts

The following result comes from similar triangles.

When two (or more) transversals cut a series of parallel lines, the ratios of their intercepts are equal.

: :AB BC DE EF

BCAB

EFDE

That is,

or

=

=

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Proof

Draw DG and EH parallel to AC .

`

EHFD;

`

`

( )

( )

( , )

( , )( )

( )

DG AB

EH BC

BCAB

EHDG

GDE HEF DG EH

DEG EFH BE CFDGE EHF

DGE

EHDG

EFDE

BCAB

EFDE

1

2

Then opposite sides of a parallelogram

Also (similarly)

corresponding s

corresponding sangle sum of s

So

From (1) and (2):

+ + +

+ + +

+ +

<

<

<

D

D

=

=

=

=

=

=

=

=

EXAMPLES

1. Find the value of x , to 3 signifi cant fi gures.

Solution

. ..

. . .

.. .

.

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 38 9 1 5

1 44

ratios of intercepts on parallel lines

#

#

=

=

=

=

^ h

CONTINUED

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2. Evaluate x and y , to 1 decimal place.

Solution

Use either similar triangles or ratios of intercepts to fi nd x . You must use similar triangles to fi nd y .

. ..

.. .

.

. .. .

.. .

.

x

x

y

y

5 8 3 42 7

3 42 7 5 8

4 6

7 1 3 42 7 3 4

3 46 1 7 1

12 7

#

#

=

=

=

=+

=

=

1. Find the value of all pronumerals, to 1 decimal place where appropriate. (a)

(b)

(c)

(d)

(e)

4.5 Exercise s

These ratios come from intercepts on parallel lines.

These ratios come from similar triangles.

Why?

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(f)

14.3

a

46c

19c

115c

46c

xc

9.125.7

8.9 y

(g)

2. Evaluate a and b to 2 decimal places.

3. Show that ABCD and CDED are similar.

4. EF bisects .GFD+ Show that DEFD and FGED are similar.

5. Show that ABCD and DEFD are similar. Hence fi nd the value of y .

4.2

4.9

6.86

1.3

5.881.82

A

C

BD

E F

yc87c

52c

6. The diagram shows two concentric circles with centre O .

Prove that (a) D .OCD;OAB <D If radius (b) . OC 5 9 cm= and

radius . OB 8 3 cm,= and the length of . CD 3 7 cm,= fi nd the length of AB , correct to 2 decimal places.

7. (a) Prove that .ADED;ABC <D Find the values of (b) x and y ,

correct to 2 decimal places.

8. ABCD is a parallelogram, with CD produced to E . Prove that .CEBD;ABF <D

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9. Show that .ABC; DAED <D Find the value of m .

10. Prove that ABCD and ACDD are similar. Hence evaluate x and y .

11. Find the values of all pronumerals, to 1 decimal place. (a)

(b)

(c)

(d)

(e)

12. Show that

(a) BCAB

FGAF

=

(b) ACAB

AGAF

=

(c) CEBD

EGDF

=

13. Evaluate a and b correct to 1 decimal place.

14. Find the value of y to 2 signifi cant fi gures.

15. Evaluate x and y correct to 2 decimal places.

ch4.indd 166 8/1/09 11:56:40 AM


Pythagoras’ Theorem

DID YOU KNOW?

The triangle with sides in the proportion 3:4:5 was known to be right angled as far back as ancient Egyptian times. Egyptian surveyors used to measure right angles by stretching out a rope with knots tied in it at regular intervals.

They used the rope for forming right angles while building and dividing fi elds into rectangular plots.

It was Pythagoras (572–495 BC)who actually discovered the relationship between the sides of the right-angled triangle. He was able to generalise the rule to all right-angled triangles.

Pythagoras was a Greek mathematician, philosopher and mystic. He founded the Pythagorean School, where mathematics, science and philosophy were studied. The school developed a brotherhood and performed secret rituals. He and his followers believed that the whole universe was based on numbers.

Pythagoras was murdered when he was 77, and the brotherhood was disbanded.

The square on the hypotenuse in any right-angled triangle is equal to the sum of the squares on the other two sides. c a b

c a b

That is,

or

2 2 2

2 2

= +

= +

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Proof

Draw CD perpendicular to AB Let ,AD x DB y= = Then x y c+ = In ADCD and ,ABCD A+ is common

D

D

;

;

( )ABC

ABC

equal corresponding s+

ADC ACB

ADC

ABAC

ACAD

cb

bx

b xcBDC

BCDB

ABBC

ay

ca

a yc

a b yc xcc y x

c c

c

90

Similarly,

Now

2

2

2 2

2

`

c+ +

<

<

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES

1. Find the value of x , correct to 2 decimal places.

Solution

c a b

x 7 449 16

65

2 2 2

2 2 2

= +

= +

= +

=

,c a b ABCIf then must be right angled2 2 2 D= +

ch4.indd 168 7/17/09 6:27:00 PM


. x 65

8 06 to 2 decimal places=

=

2. Find the exact value of y .

Solution

c a b

y

y

y

y

8 4

64 16

48

48

16 3

4 3

2 2 2

2 2 2

2

2

`

#

= +

= +

= +

=

=

=

=

3. Find the length of the diagonal in a square with sides 6 cm. Answer to 1 decimal place.

Solution

6 cm

6 cm

.

c a b

c

6 672

728 5

2 2 2

2 2

= +

= +

=

=

=

So the length of the diagonal is 8.5 cm.

Leave the answer in surd form for the exact

answer.

CONTINUED

ch4.indd 169 7/17/09 6:27:03 PM


1. Find the value of all pronumerals, correct to 1 decimal place. (a)

(b)

(c)

(d)

2. Find the exact value of all pronumerals. (a)

(b)

(c)

(d)

4.6 Exercises

4. A triangle has sides 5.1 cm, 6.8 cm and 8.5 cm. Prove that the triangle is right angled.

Solution

6.8 cm

8.5 cm5.1 cm

Let .c 8 5= (largest side) and a and b the other two smaller sides.

. . .

. .

a b

c

c a b

5 1 6 872 25

8 572 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +

So the triangle is right angled .

ch4.indd 170 7/17/09 6:27:10 PM


3. Find the slant height s of a cone with diameter 6.8 m and perpendicular height 5.2 m, to 1 decimal place.

4. Find the length of CE , correct to 1 decimal place, in this rectangular pyramid. 8.6 AB cm= and 15.9 .CF cm=

5. Prove that ABCD is a right-angled triangle.

6. Show that XYZD is a right-angled isosceles triangle.

X

Y Z1

1 2

7. Show that .AC BC2=

8. (a) Find the length of diagonal AC in the fi gure.

Hence, or otherwise, prove (b) that AC is perpendicular to DC .

9. Find the length of side AB in terms of b .

10. Find the exact ratio of YZXY in

terms of x and y in .XYZD

ch4.indd 171 7/17/09 6:27:15 PM


11. Show that the distance squared between A and B is given by .d t t13 180 6252 2= - +

12. An 850 mm by 1200 mm gate is to have a diagonal timber brace to give it strength. To what length should the timber be cut, to the nearest mm?

13. A rectangular park has a length of 620 m and a width of 287 m. If I walk diagonally across the park, how far do I walk?

14. The triangular garden bed below is to have a border around it. How many metres of border are needed, to 1 decimal place?

15. What is the longest length of stick that will fi t into the box below, to 1 decimal place?

16. A ramp is 4.5 m long and 1.3 m high. How far along the ground does the ramp go? Answer correct to one decimal place .

4.5 m1.3 m

17. The diagonal of a television screen is 72 cm. If the screen is 58 cm high, how wide is it?

18. A property has one side 1.3 km and another 1.1 km as shown with a straight road diagonally through the middle of the property. If the road is 1.5 km long, show that the property is not rectangular.

1.3 km

1.1 km

1.5 km

19. Jodie buys a ladder 2 m long and wants to take it home in the boot of her car. If the boot is 1.2 m by 0.7 m, will the ladder fi t?

ch4.indd 172 7/17/09 6:27:24 PM


Types of Quadrilaterals

A quadrilateral is any four-sided fi gure

In any quadrilateral the sum of the interior angles is 360c

20. A chord AB in a circle with centre O and radius 6 cm has a perpendicular line OC as shown 4 cm long.

A

B

O

C

6 cm4 cm

By fi nding the lengths of (a) AC and BC , show that OC bisects the chord .

By proving congruent (b) triangles, show that OC bisects the chord .

Proof

Draw in diagonal AC

180 ( )( )

,

ADC DCA CAD

ABC BCA CAB

ADC DCA CAD ABC BCA CAB

ADC DCB CBA BAD

180

360

360

angle sum ofsimilarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =

ch4.indd 173 7/17/09 6:27:32 PM


opposite sides• of a parallelogram are equal • opposite angles of a parallelogram are equal • diagonals in a parallelogram bisect each other each diagonal bisects the parallelogram into two • congruent triangles

A quadrilateral is a parallelogram if: both pairs of • opposite sides are equal both pairs of • opposite angles are equal one • pair of sides is both equal and parallel the • diagonals bisect each other

These properties can all be proven.

Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel

EXAMPLE

Find the value of .i

Solution

120 56 90 360

266 360

94

angle sum of quadrilaterali

i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS

ch4.indd 174 7/17/09 6:27:36 PM


Rhombus

A rectangle is a parallelogram with one angle a right angle

the same as for a parallelogram, and also • diagonals are equal •

A quadrilateral is a rectangle if its diagonals are equal

Application

Builders use the property of equal diagonals to check if a rectangle is accurate. For example, a timber frame may look rectangular, but may be slightly slanting. Checking the diagonals makes sure that a building does not end up like the

Leaning Tower of Pisa!

It can be proved that all sides are equal.

If one angle is a right angle, then you can prove all angles are

right angles.

A rhombus is a parallelogram with a pair of adjacent sides equal

the same as for parallelogram, and also • diagonals bisect at right angles • diagonals bisect the angles of the rhombus •

Rectangle

PROPERTIES

PROPERTIES

TEST

ch4.indd 175 7/31/09 4:25:28 PM


Square

A square is a rectangle with a pair of adjacent sides equal

• the same as for rectangle, and also diagonals are perpendicular • diagonals make angles of • 45c with the sides

Trapezium

A trapezium is a quadrilateral with one pair of sides parallel

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal

A quadrilateral is a rhombus if: all sides are equal • diagonals bisect each other at right angles •

TESTS

PROPERTIES

ch4.indd 176 7/17/09 6:27:44 PM


EXAMPLES

1. Find the values of ,i x and y , giving reasons.

Solution

( )

. ( )

. ( )

x

y

83

6 7

2 3

opposite s in gram

cm opposite sides in gram

cm opposite sides in gram

c + <

<

<

i =

=

=

2. Find the length of AB in square ABCD as a surd in its simplest form if 6 .BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let

Since is a square, adjacent sides equal

Also, by definitionc+

=

= =

=

By Pythagoras’ theorem:

3

c a b

x x

x

x

x

6

36 2

18

18

2 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED

ch4.indd 177 7/17/09 6:27:47 PM


1. Find the value of all pronumerals, giving reasons. (a)

(b)

(c)

(d)

(e)

(f)

(g)

4.7 Exercises

3. Two equal circles have centres (a) O and P respectively. Prove that OAPB

is a rhombus. Hence, or otherwise, show that (b) AB is the perpendicular bisector

of OP .

Solution

(a) ( )

( )

OA OB

PA PB

OA OB PA PB

equal radii

similarly

Since the circles are equal,

=

=

= = =

since all sides are equal, OAPB is a rhombus The diagonals in any rhombus are perpendicular bisectors. (b)

Since OAPB is a rhombus, with diagonals AB and OP , AB is the perpendicular bisector of OP .

ch4.indd 178 7/17/09 6:27:51 PM


2. Given ,AB AE= prove CD is perpendicular to AD .

3. (a) Show that C xc+ = and ( ) .B D x180 c+ += = -

Hence show that the sum of (b) angles of ABCD is .360c

4. Find the value of a and b .

5. Find the values of all pronumerals, giving reasons.

(a)

(b)

(c)

(d)

(e)

7

y3x

x+6

(f)

6. In the fi gure, BD bisects .ADC+ Prove BD also bisects .ABC+

7. Prove that each fi gure is a parallelogram. (a)

(b)

ch4.indd 179 7/17/09 6:27:55 PM


(c)

(d)

8. Evaluate all pronumerals.

(a)

(b)

ABCD is a kite

(c)

(d)

(e)

9. The diagonals of a rhombus are 8 cm and 10 cm long. Find the length of the sides of the rhombus.

10. ABCD is a rectangle with .EBC 59c+ = Find ,ECB EDC+ + and .ADE+

11. The diagonals of a square are 8 cm long. Find the exact length of the side of the square.

12. In the rhombus, .ECB 33c+ = Find the value of x and y .

Polygons

A polygon is a closed plane fi gure with straight sides

A regular polygon has all sides and all interior angles equal

ch4.indd 180 7/17/09 6:28:01 PM


Proof

Draw any n -sided polygon and divide it into n triangles as shown. Then the total sum of angles is n 180# c or 180 .n But this sum includes all the angles at O . So the sum of interior angles is 180 360 .n c- That is, S n

n

180 360

2 180# c

= -

= -] g

EXAMPLES

4-sided (square)

3-sided (equilateral

triangle)

5-sided (pentagon)

6-sided (hexagon)

8-sided (octagon)

10-sided (decagon)

DID YOU KNOW?

Carl Gauss (1777–1855) was a famous German mathematician, physicist and astronomer. When he was 19 years old, he showed that a 17-sided polygon could be constructed using a ruler and compasses. This was a major achievement in geometry.

Gauss made a huge contribution to the study of mathematics and science, including correctly calculating where the magnetic south pole is and designing a lens to correct astigmatism.

He was the director of the Göttingen Observatory for 40 years. It is said that he did not become a professor of mathematics because he did not like teaching.

The sum of the interior angles of an n -sided polygon is given by

( 2) 180

S n

S n

180 360

or # c

= -

= -

The sum of the exterior angles of any polygon is 360c

Proof

Draw any n -sided polygon. Then the sum of both the exterior and interior angles is .n 180# c

n

n nn n

180

180 180 360180 180 360

360

Sum of exterior angles sum of interior angles# c

c

c

c

= -

= - -

= - +

=

] g

ch4.indd 181 7/17/09 6:28:08 PM


EXAMPLES

1. Find the sum of the interior angles of a regular polygon with 15 sides. How large is each angle?

Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

#

#

#

c

c

c

c

=

= -

= -

=

=

Each angle has size .2340 15 156'c c=

2. Find the number of sides in a regular polygon whose interior angles are .140c

Solution

Let n be the number of sides Then the sum of interior angles is 140n

( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

#

#

c

c

= -

= -

= -

=

=

So the polygon has 9 sides.

There are n sides and so n angles, each 140 .c

1. Find the sum of the interior angles of

a pentagon (a) a hexagon (b) an octagon (c) a decagon (d) a 12-sided polygon (e) an 18-sided polygon (f)

2. Find the size of each interior angle of a regular

pentagon (a) octagon (b) 12-sided polygon (c) 20-sided polygon (d) 15-sided polygon (e)

3. Find the size of each exterior angle of a regular

hexagon (a) decagon (b) octagon (c) 15-sided polygon (d)

4. Calculate the size of each interior angle in a regular 7-sided polygon, to the nearest minute.

5. The sum of the interior angles of a regular polygon is .1980c

How many sides has the (a) polygon?

Find the size of each interior (b) angle, to the nearest minute.

4.8 Exercises

ch4.indd 182 7/17/09 6:28:12 PM


6. Find the number of sides of a regular polygon whose interior angles are .157 30c l

7. Find the sum of the interior angles of a regular polygon whose exterior angles are .18c

8. A regular polygon has interior angles of .156c Find the sum of its interior angles.

9. Find the size of each interior angle in a regular polygon if the sum of the interior angles is .5220c

10. Show that there is no regular polygon with interior angles of .145c

11. Find the number of sides of a regular polygon with exterior angles

(a) 40c (b) 03 c (c) 45c (d) 36c (e) 12c

12. ABCDEF is a regular hexagon.

F

E D

A B

C

Show that triangles (a) AFE and BCD are congruent .

Show that (b) AE and BD are parallel .

13. A regular octagon has a quadrilateral ACEG inscribed as shown.

D

A

B

E

C

F

G

H

Show that ACEG is a square .

14. In the regular pentagon below, show that EAC is an isosceles triangle .

D

A

BE

C

15. (a) Find the size of each exterior angle in a regular polygon with side p .

Hence show that each interior (b)

angle is ( )pp180 2-

.

ch4.indd 183 7/17/09 6:28:15 PM


Areas

Most areas of plane fi gures come from the area of a rectangle.

Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b

Draw rectangle ABCD , where b length= and h breadth= .

A square is a special rectangle.

The area of a triangle is half the area of a rectangle.

ch4.indd 184 7/17/09 6:28:18 PM


bharea

21

21

21

21

` =

DEF AEFD CEF EBCFArea area and area areaD D= =

CDE ABCDarea ` D =

A bhThat is, =

area

A bh=

Proof

In parallelogram ABCD , produce DC to E and draw BE perpendicular to CE . Then ABEF is a rectangle.

Area ABEF bh= In ADFD and ,BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90

opposite sides of a rectangle

opposite sides of a parallelogram

by RHS,

area areaSo area area

`

`

c+ +

/D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a parallelogram is the same as the area of

two triangles.

A xy21

=

( x and y are lengths of diagonals)

Parallelogram

ch4.indd 185 8/7/09 12:57:48 PM


( )A h a b21

= +

Proof

DE x

DF x a

FC b x ab x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof

Let AC x= and BD y= By properties of a rhombus,

AE EC x21

= = and DE EB y21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area

total area of rhombus

21

21

41

21

21

41

41

41

21

:

:

`

D

D

=

=

=

=

= +

=

Trapezium

ch4.indd 186 7/17/09 6:28:22 PM


A r2r=

EXAMPLES

1. Find the area of this trapezium.

Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

#

= +

= +

=

=

2. Find the area of the shaded region in this fi gure.

8.9

cm

3.7

cm

12.1 cm

4.2 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2

Area trapezium area area rectangle area

21

21

21

21

D D= + +

= + + - -

= + + - -

= +

Circle

ch4.indd 187 7/31/09 4:25:28 PM


Solution

. .

.

. .

. . .

.

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54107 69 15 54

92 15

Area large rectangle

cmArea small rectangle

cmshaded area

cm

2

2

2

#

#

`

=

=

=

=

=

=

= -

=

3 . A park with straight sides of length 126 m and width 54 m has semi-circular ends as shown. Find its area, correct to 2 decimal places.

126 m

54 m

Solution

-Area of 2 semi circles area of 1 circle=

2

( )

.

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

.

.

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

#=

=

= +

=

1. Find the area of each fi gure. (a)

(b)

4.9 Exercises

ch4.indd 188 7/17/09 6:28:28 PM


(c)

(d)

(e)

(f)

(g)

2. Find the area of a rhombus with diagonals 2.3 m and 4.2 m.

3. Find each shaded area .(a)

(b)

(c)

(d)

(e)

6 cm

2 cm

4. Find the area of each fi gure. (a)

(b)

ch4.indd 189 7/17/09 6:28:31 PM


(c)

(d)

(e)

5. Find the exact area of the fi gure.

6. Find the area of this fi gure, correct to 4 signifi cant fi gures. The arch is a semicircle.

7. Jenny buys tiles for the fl oor of her bathroom (shown top next column) at $45.50 per .m2 How much do they cost altogether?

8. The dimensions of a battleaxe block of land are shown below.

Find its area. (a) A house in the district where (b)

this land is can only take up 55% of the land. How large (to the nearest m2 ) can the area of the house be?

If the house is to be a (c) rectangular shape with width 8.5 m, what will its length be?

9. A rhombus has one diagonal 25 cm long and its area is 600 .cm2 Find the length of

its other diagonal and (a) its side, to the nearest cm. (b)

10. The width w of a rectangle is a quarter the size of its length. If the width is increased by 3 units while the length remains constant, fi nd the amount of increase in its area in terms of w .

ch4.indd 190 7/17/09 6:28:34 PM


Test Yourself 4

The perimeter is the distance around the outside of the fi gure.

1. Find the values of all pronumerals (a)

(b)

(c)

x(d)

(O is the centreof the circle.)

(e)

(f)

(g)

2. Prove that AB and CD are parallel lines.

3. Find the area of the fi gure, to 2 decimal places.

4. (a) Prove that triangles ABC and ADE are similar.

Evaluate (b) x and y to 1 decimal place.

5. Find the size of each interior angle in a regular 20-sided polygon.

6. Find the volume of a cylinder with radius 5.7 cm and height 10 cm, correct to 1 decimal place.

7. Find the perimeter of the triangle below.

ch4.indd 191 7/17/09 6:28:40 PM


8. (a) Prove triangles ABC and ADC are congruent in the kite below.

Prove triangle (b) AOB and COD are congruent. ( O is the centre of the circle.)

9. Find the area of the fi gure below.

10. Prove triangle ABC is right angled.

11. Prove .AGAF

ACAB

=

12. Triangle ABC is isosceles, and AD bisects BC .

Prove triangles (a) ABD and ACD are congruent.

Prove (b) AD and BC are perpendicular.

13. Triangle ABC is isosceles, with .AB AC= Show that triangle ACD is isosceles.

14. Prove that opposite sides in any parallelogram are equal.

15. A rhombus has diagonals 6 cm and 8 cm. Find the area of the rhombus. (a) Find the length of its side. (b)

16. The interior angles in a regular polygon are .140c How many sides has the polygon?

17. Prove AB and CD are parallel.

ch4.indd 192 7/17/09 6:28:51 PM


18. Find the area of the fi gure below.

10 cm

2 cm

5 cm

6 cm

8 cm

19. Prove that z x y= + in the triangle below.

20. (a) Prove triangles ABC and DEF are similar.

Evaluate (b) x to 1 decimal place.

1. Find the value of x .

2. Evaluate x , y and z .

3. Find the sum of the interior angles of a regular 11-sided polygon. How large is each exterior angle?

4. Given ,BAD DBC+ += show that ABDD and BCDD are similar and hence fi nd d .

5. Prove that ABCD is a parallelogram. .AB DC=

6. Find the shaded area.

Challenge Exercise 4

ch4.indd 193 7/17/09 6:29:02 PM


7. Prove that the diagonals in a square make angles of 45c with the sides.

8. Prove that the diagonals in a kite are perpendicular.

9. Prove that MN is parallel to XY .

10. Evaluate x .

11. The letter Z is painted on a billboard.

Find the area of the letter. (a) Find the exact perimeter of the letter. (b)

12. Find the values of x and y correct to 1 decimal place.

13. Find the values of x and y , correct to 2 decimal places.

14. ABCD is a square and BD is produced to

E such that .DE BD21

=

Show that (a) ABCE is a kite.

Prove that (b) DEx

22

= units when

sides of the square are x units long.

ch4.indd 194 7/17/09 6:29:12 PM

Documents

Maths in Focus - Margaret Grove - ch4