maths in focus - margaret grove - ch4
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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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139Chapter 4 Geometry 1
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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140 Maths In Focus Mathematics Preliminary Course
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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141Chapter 4 Geometry 1
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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142 Maths In Focus Mathematics Preliminary Course
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,
vertically opposite angles
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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143Chapter 4 Geometry 1
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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144 Maths In Focus Mathematics Preliminary Course
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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145Chapter 4 Geometry 1
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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146 Maths In Focus Mathematics Preliminary Course
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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147Chapter 4 Geometry 1
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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148 Maths In Focus Mathematics Preliminary Course
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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149Chapter 4 Geometry 1
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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150 Maths In Focus Mathematics Preliminary Course
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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151Chapter 4 Geometry 1
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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152 Maths In Focus Mathematics Preliminary Course
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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153Chapter 4 Geometry 1
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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154 Maths In Focus Mathematics Preliminary Course
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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155Chapter 4 Geometry 1
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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156 Maths In Focus Mathematics Preliminary Course
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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157Chapter 4 Geometry 1
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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158 Maths In Focus Mathematics Preliminary Course
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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159Chapter 4 Geometry 1
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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160 Maths In Focus Mathematics Preliminary Course
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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161Chapter 4 Geometry 1
.. .
.
. ..
. . .
.. .
.
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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162 Maths In Focus Mathematics Preliminary Course
. .
.
. .. . .
.. .
.
.
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
vertically opposite angles
`
+ +
= =
= =
=
=
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
=
=
ch4.indd 162 7/17/09 6:26:12 PM
163Chapter 4 Geometry 1
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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164 Maths In Focus Mathematics Preliminary Course
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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165Chapter 4 Geometry 1
(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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166 Maths In Focus Mathematics Preliminary Course
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
167Chapter 4 Geometry 1
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
= +
= +
ch4.indd 167 7/17/09 6:26:40 PM
168 Maths In Focus Mathematics Preliminary Course
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
169Chapter 4 Geometry 1
. 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
170 Maths In Focus Mathematics Preliminary Course
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
171Chapter 4 Geometry 1
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
172 Maths In Focus Mathematics Preliminary Course
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
173Chapter 4 Geometry 1
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
174 Maths In Focus Mathematics Preliminary Course
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
175Chapter 4 Geometry 1
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
176 Maths In Focus Mathematics Preliminary Course
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
177Chapter 4 Geometry 1
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
178 Maths In Focus Mathematics Preliminary Course
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
179Chapter 4 Geometry 1
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
180 Maths In Focus Mathematics Preliminary Course
(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
181Chapter 4 Geometry 1
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
182 Maths In Focus Mathematics Preliminary Course
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
183Chapter 4 Geometry 1
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
184 Maths In Focus Mathematics Preliminary Course
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
185Chapter 4 Geometry 1
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
186 Maths In Focus Mathematics Preliminary Course
( )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
187Chapter 4 Geometry 1
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
188 Maths In Focus Mathematics Preliminary Course
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
189Chapter 4 Geometry 1
(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
190 Maths In Focus Mathematics Preliminary Course
(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
191Chapter 4 Geometry 1
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
192 Maths In Focus Mathematics Preliminary Course
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
193Chapter 4 Geometry 1
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
194 Maths In Focus Mathematics Preliminary Course
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
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