chapter · than mine”. jane used a ruler to measure the sides of each slice. “see, my slice has...
TRANSCRIPT
17Chapter 17
Contents:
A TransformationsB Congruent figures
C Congruent triangles
D Proof using congruence
Congruence and
transformations
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Opening problem
352 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
Jane cut two triangular slices of cheesecake, and gave
one to her brother Nathan.
“That’s not fair”, Nathan said, “your slice is bigger
than mine”.
Jane used a ruler to measure the sides of each slice.
“See, my slice has sides 5 cm, 6 cm, and 7 cm, and so
does yours. That means the slices are the same size.”
“Not necessarily”, said Nathan, “the slices might have
the same sides, but the angles might be different”.
Things to think about:
a Who do you think is correct?
b What mathematical argument can you use to justify your answer?
Congruence is a branch of geometry that deals with objects which are identical in size and shape.
In this chapter we will review transformations, and look at how we can use transformations to
define congruence. We will then use congruence to prove the properties of polygons we have
studied earlier in the year.
A transformation is a process which changes either the size, shape, orientation, or position of a
figure.
When we perform a transformation, the original figure is called the object, and the resulting figure
is called the image.
In this section we revise the translation, reflection, and rotation transformations.
TRANSLATIONS
A translation is a transformation in which every point on the figure moves a fixed distance in
a given direction.
This object has been translated 4 units right and 3 units
down.
TRANSFORMATIONSA
Under a translation, the size
and shape of an object does
not change. Only the position
of the object changes.
4
¡3object
image
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Y:\HAESE\AUS_08\AUS08_17\352AUS08_17.cdr Tuesday, 2 August 2011 11:03:12 AM BEN
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 353
We can also translate objects plotted on a Cartesian plane.
Self Tutor A0 is the image of
the object A.Translate this object 5 units
left and 2 units up.
Example 1
2
¡5
A
A0
A
Self Tutor
a Translate the quadrilateral ABCD 2 units
right and 4 units up.
b State the vertex coordinates of the image
quadrilateral.
a b The vertices of the image quadrilateral
are A0(¡1, 3), B0(1, 1), C0(1, ¡1),
and D0(¡1, ¡1).
EXERCISE 17A.1
1 Translate the given figures in the direction indicated:
a
3 units right,
2 units down.
b
4 units right.
c
3 units down.
Example 2
y
xA
D C
B
y
xA
D
B
A
D C
B
0
0
00
C
4
2
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Y:\HAESE\AUS_08\AUS08_17\353AUS08_17.cdr Wednesday, 17 August 2011 3:43:12 PM BEN
354 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
d
3 units left,
1 unit up.
e
1 unit left,
4 units down.
f
3 units left,
3 units up.
2 Determine the translation from A to A0 in the following:
a b c
d e f
3 An object A is translated 5 units right and 3 units down to A0. Describe the translation from
A0 back to A.
4 Translate the following figures in the direction given, and state the vertex coordinates of the
image:
a
Translate 4 units left.
b
Translate 2 units right,
then 1 unit down.
c
Translate 3 units left,
then 4 units up.
a
y
x
A
C
B
y
x
A B
D C
y
xA B
CD
EF
A A0
A
A0
A
A0
A
A0
A
A0
A
A0
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Y:\HAESE\AUS_08\AUS08_17\354AUS08_17.cdr Tuesday, 16 August 2011 12:56:00 PM BEN
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 355
Self Tutor
Is this transformation a translation?
In a translation, every point
on the figure moves the
same distance in the same
direction.
However, A has moved 3units right and 1 unit up,
while C has moved 4 units
right and 1 unit up.
So, this transformation is
not a translation.
5 Are the following transformations translations? If so, describe the translation.
a b c
6 Consider the figures alongside.
a Which of these figures is a translation
of figure C? Describe the translation
from figure C to this figure.
b Which of these figures is a translation
of figure G? Describe the translation
from figure G to this figure.
c Which of these figures cannot be
translated to any other figure?
Example 3
The triangles do
not have the same
shape, so it cannot
be a translation.
A
C
B
B
A
C
0
0
0
y
x
A
B
C
D
E
F
G
H
I
A
C
B
B
A
C
0
0
0
A B
CD
A B0 0
D C0 0
A B
CD A0 B0
C0D0
A B
C D
EF
A0 B0
C0 D0
E 0F 0
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Y:\HAESE\AUS_08\AUS08_17\355AUS08_17.cdr Wednesday, 17 August 2011 11:00:17 AM BEN
Investigation 1 Reflections
356 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
REFLECTIONS
Consider the figure alongside. The object has been
reflected in the mirror line to form its image. In
this case we might call it the mirror image.
You will need: A mirror, paper, pencil, ruler.
What to do:
1 Make two copies of the figures shown below:
a b
c d
2 Put the mirror along the mirror line m on one copy. What do you notice in the mirror?
3 Draw the reflection as accurately as you can on the second copy.
4 Cut out the second copy with its reflection and fold it along the mirror line. You should
find that the two parts of the figure can be folded exactly onto one another along the
mirror line.
When point A is reflected in a mirror line, A and its image
A0 are the same distance from the mirror line, and the line
joining A and A0 is perpendicular to the mirror line.
mirror line
object image
PRINTABLE
FIGURES
m
m
m
mirror line
A0A
m
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CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 357
Self Tutor
Reflect the following figures in the given mirror lines:
a b
a b
EXERCISE 17A.2
1 Copy the following figures onto grid paper and reflect them in the given mirror
lines:
a b c
d e f
Self Tutor
Reflect this figure in the y-axis.
Example 5
Example 4
m
A
A A0
m
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DIAGRAMS
m
m m
mm
m
y
xA
y
xA 0A
m
A
m
A0
A
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Y:\HAESE\AUS_08\AUS08_17\357AUS08_17.cdr Friday, 26 August 2011 8:46:39 AM BEN
358 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
2 Reflect these figures in the x-axis:
a b c
3 Reflect these figures in the y-axis:
a b c
4 A quadrilateral has vertices A(1, 5), B(4, 3), C(4, 1), and D(1, 1).
a Plot the quadrilateral ABCD on a Cartesian plane.
b Reflect ABCD in the x-axis, and state the vertex coordinates of the image.
c Reflect ABCD in the y-axis, and state the vertex coordinates of the image.
5 Copy and complete:
a When the point (a, b) is reflected in the x-axis, the image has coordinates (::::, ::::).
b When the point (a, b) is reflected in the y-axis, the image has coordinates (::::, ::::).
6 For each of the following, determine whether A0 is a reflection of A in the x-axis:
a b c
7 a Which two of the figures are reflections of
each other?
b Which of the axes is the mirror line for this
reflection?
c Which two of the figures are translations
of each other?
y
x
y
x
y
x
y
x
y
x
y
x
y
x
A
A0
y
x
A
A0
y
x
A B
DC
y
x
A
A0
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Y:\HAESE\AUS_08\AUS08_17\358AUS08_17.cdr Tuesday, 16 August 2011 12:59:33 PM BEN
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 359
ROTATIONS
When a wheel moves about its axle, we say that the wheel
rotates.
The centre point on the axle is the centre of rotation.
The angle through which the wheel turns is the angle of
rotation.
Other examples of rotation are the movement of the hands of a clock, and opening and closing a
door.
A rotation is a transformation in which every point on the figure is turned through a given angle
about a fixed point.
The fixed point is called the centre of rotation and is usually labelled O.
For example, the object alongside has been rotated 90±
anticlockwise about O.
When a point is rotated about O, that point and its image
are the same distance from O.
OA = OA0, OB = OB0, and so on.
Self Tutor
Rotate the given figures about O through the angle indicated:
a
180± clockwise
b
90± anticlockwise
c
90± clockwise
a b c
Example 6
O
90°
object
image
D0
A B
C
DEC0
B0
A0 E0We draw circle
arcs centred at O
to make sure that a
point and its image
are the same
distance from O.
O
O
O
O
O
O
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Y:\HAESE\AUS_08\AUS08_18 - new 17\359AUS08_17.cdr Monday, 1 August 2011 9:50:28 AM BEN
360 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
EXERCISE 17A.3
1 Derek is sitting in chair A. Which chair will
he move to if he rotates anticlockwise about O
through an angle of:
a 90± b 270± c 180±?
2 Rotate the given figures about O through the angle indicated:
a
90± anticlockwise
b
180±
c
90± clockwise
Self Tutor
a State the vertex coordinates of triangle ABC.
b Rotate the triangle 90± clockwise about the
origin O.
c State the vertex coordinates of the image.
a The triangle has vertices A(¡2, 4),
B(¡1, 4), and C(¡2, 1).
c The image triangle has vertices A0(4, 2),
B0(4, 1), and C0(1, 2).
b
Example 7
O
A
BD
C
We rotate
anticlockwise
unless we are
told otherwise.
O
O
y
x
A B
C
y
x
A B
C
0A0C
0B
O
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Y:\HAESE\AUS_08\AUS08_17\360AUS08_17.cdr Thursday, 18 August 2011 10:15:07 AM BEN
Discussion
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 361
3 a State the vertex coordinates of triangle PQR.
b Rotate the triangle 90± anticlockwise about the
origin O.
c State the vertex coordinates of the image.
4 A quadrilateral has vertices A(¡4, ¡3), B(¡1, ¡3), C(¡1, ¡4), and D(¡4, ¡4).
a Plot ABCD on a Cartesian plane.
b Rotate ABCD 90± clockwise about the origin O.
c State the vertex coordinates of the image.
5 a Which of the figures alongside is a rotation of A
about the origin?
b Determine the angle of rotation from figure A to
this figure.
When an object is translated, reflected, or rotated, does the size of the object change?
Does the shape of the object change?
Two figures are congruent if they are identical in size and shape. They do not need to have the
same orientation.
For example, the figures alongside are congruent
even though one is a rotation of the other.
CONGRUENT FIGURESB
y
x
P
Q
R
y
x
AB
CD
congruent figures
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Y:\HAESE\AUS_08\AUS08_18 - new 17\361AUS08_17.cdr Monday, 1 August 2011 10:02:22 AM BEN
Activity Creating congruent figures
362 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
You will need: Two sheets of paper, scissors.
What to do:
1 Draw a shape on one of the sheets of paper.
2 Place the second sheet of paper behind it, and hold them together tightly. Carefully cut
out the shape, cutting through both sheets of paper. This will give you two congruent
figures.
3 In a group or as a class, place both figures from
each student in a box, and mix the figures up.
4 Try to pair up the congruent figures. How can you
tell that two figures are congruent?
DEMO
The figures alongside are congruent. The
corresponding sides and angles in the
figures are identical. If we were to place
one figure on top of the other, they would
match each other perfectly.
Self Tutor
Are the following pairs of figures congruent?
a b c
a The figures do not have the same shape, so they are not congruent.
b The figures are identical in size and shape even though one is rotated. They are therefore
congruent.
c Although the figures have the same shape, they are not the same size. They are not
congruent.
Example 8
DEMO
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Y:\HAESE\AUS_08\AUS08_18 - new 17\362AUS08_17.cdr Monday, 1 August 2011 10:04:19 AM BEN
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 363
EXERCISE 17B.1
1 Are the following pairs of figures congruent?
a b
c d
2 Which two of these figures are congruent?
A B C D E
3 Which three of these figures are congruent?
A B C D E
4 Quadrilaterals EFGH and ABCD are
congruent.
Determine the:
a length of side [EF]
b size of angle
c perimeter of EFGH.
USING TRANSFORMATIONS TO DEFINE CONGRUENCE
The figures A and B alongside are
congruent. If we translate figure A
to figure B, the two figures fit together
perfectly.
EF
G
HA
B
CD
90° 98°
107°
65°
12 cm
10 cm
7.8 cm
8 cm
DEMO
A
B
FbGH
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Y:\HAESE\AUS_08\AUS08_17\363AUS08_17.cdr Monday, 22 August 2011 10:46:40 AM BEN
364 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
The figures P and Q are congruent.
They have different orientations, but
they still have the same size and shape.
We can see this by rotating P so that
it has the same orientation as Q, then
translating the image P0 to Q.
We can therefore use transformations to define congruence:
Two figures are congruent if one figure lies exactly on top of the other after a combination of
translations, rotations, and reflections.
Self Tutor
Show that A and B are congruent by
transforming A onto B.
We first reflect figure A in the x-axis. We
then translate A0 5 units right and 1 unit
down.
The image fits onto figure B exactly, so A
and B are congruent.
Example 9
DEMO
P
Q
QP0
The orientation of
a figure refers to
the direction it is
facing.
DEMO
X Y
X Y
m
X0
y
x
A
B
y
x
A
B
A0
The figures X and Y are also
congruent. To see that they are the
same size and shape, we reflect figure
X in a mirror line, then translate the
image X0 to Y.
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Y:\HAESE\AUS_08\AUS08_17\364AUS08_17.cdr Wednesday, 17 August 2011 11:03:43 AM BEN
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 365
EXERCISE 17B.2
1 Show that A is congruent to B by transforming A to B in a single transformation:
a b c
d e f
2 Show that A is congruent to B by transforming A to B in a combination of transformations:
a b
c d
e f
y
x
A
B
y
x
A
B
y
x
A
B
y
x
A
B
y
xA B
y
x
A
B
PRINTABLE
DIAGRAMS
First reflect or rotate A
so the figures have the
same orientation. Then
translate if required.
y
xA
B
y
x
B
A
y
x
B
A
y
x
A
B
y
x
A
By
x
A
B
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Y:\HAESE\AUS_08\AUS08_17\365AUS08_17.cdr Monday, 22 August 2011 10:47:28 AM BEN
Investigation 2 Congruent triangles
366 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
3 Are the following pairs of figures congruent?
a b c
The triangles alongside have identical side
lengths and angles, so the triangles are
congruent.
However, we do not need to know all of
the information given to conclude that the
triangles are congruent.
For example, as the investigation below demonstrates, knowing that the triangles have the same
side lengths is sufficient to conclude that the triangles are congruent.
What to do:
1 Click on the icon to run the computer demonstration.
The computer will generate a triangle with side lengths 8 cm, 10 cm, and
12 cm. You will now make another triangle with these dimensions.
2 Choose which of the side lengths you would like to start
with. For example, you may choose to start with the
12 cm side.
3 Drag the third vertex of the triangle around until the
remaining two sides have the correct lengths.
4 Compare your triangle with the one generated by the computer. Watch the transformations
and decide if the triangles are congruent.
5 Construct another triangle with side lengths 8 cm, 10 cm, and 12 cm, and test this triangle
for congruence with the other two.
CONGRUENT TRIANGLESC
y
x
5 cm
8 cm
7 cm
60° 38°
82°
5 cm
7 cm
8 cm60°
38°
82°
DEMO
12 cm
12 cm
10 cm8 cm
y
x
y
x
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Y:\HAESE\AUS_08\AUS08_17\366AUS08_17.cdr Monday, 22 August 2011 10:49:46 AM BEN
Investigation 3 Constructing triangles
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 367
In this investigation we will discover other conditions which allow us to conclude that two
triangles are congruent.
You will need: Paper, ruler, protractor.
What to do:
1 Two sides and an included angle
Draw a triangle with two side lengths 8 cm
and 12 cm, with an angle of 25± between
these sides. How many different triangles
can be constructed?
2 Two angles and a corresponding side
Draw a triangle with two angles measuring
70± and 45±, with the side between these
angles being 10 cm long. How many
different triangles can be constructed?
4 Two sides and a non-included angle
Draw a triangle with two side lengths
8 cm and 12 cm, with an angle of 25±
between the 12 cm side and the third side as
shown. How many different triangles can
be constructed?
5 Three angles
Draw a triangle with angles 50±, 60±, and
70±. How many different triangles can be
constructed?
12 cm
8 cm
25°
10 cm
70° 45°
10 cm6 cm
The hypotenuse is
the longest side of
a right angled
triangle.
12 cm
8 cm
25°
60° 50°
70°
You should have found that your triangles
and the computer’s triangles were congruent.
If we know that two triangles have the same
side lengths, then these triangles must be
congruent.
3 Right angle, hypotenuse, and a side
Draw a right angled triangle with
hypotenuse 10 cm, and one other side
6 cm long. How many different triangles
can be constructed?
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Y:\HAESE\AUS_08\AUS08_17\367AUS08_17.cdr Wednesday, 17 August 2011 11:10:18 AM BEN
368 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
You should have made the following discoveries:
Two triangles are congruent if any one of the following is true:
² All corresponding sides are equal in length. (SSS)
² Two sides and the included angle are equal. (SAS)
² Two angles and a pair of corresponding sides are
equal. (AAcorS)
² For right angled triangles, the hypotenuses and one
pair of sides are equal. (RHS)
We usually indicate our reason why two triangles are congruent by writing one of the abbreviations
given above in bold.
If we know two side lengths and a non-included angle, there may be two ways to construct the
triangle. This is therefore not sufficient information to show that two triangles are congruent.
If we know all angles of a triangle, the triangle may still vary in size. This is therefore not sufficient
information to show that two triangles are congruent.
Self Tutor
Are these pairs of triangles congruent? Give reasons for your answers.
a b
c d
a Yes fRHSg b Yes fSASg
c No. This is not AAcorS as the equal sides are not in corresponding positions. One is
opposite angle ®, the other is opposite angle ¯.
d Yes fAAcorSg
Example 10
� �
� �
4 cm
�
�
4 cm
� �
��
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Y:\HAESE\AUS_08\AUS08_17\368AUS08_17.cdr Wednesday, 17 August 2011 11:10:45 AM BEN
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 369
Once we have established that two triangles are congruent, we can deduce that the remaining
corresponding sides and angles of the triangles are equal.
Self Tutor
Consider the two triangles alongside.
a Show that the triangles are
congruent.
b What can be deduced from this
congruence?
a AB = XY, BC = YZ
and AbBC = XbYZ
So, 4ABC »= 4XYZ fSASg
b AC = XZ
BbAC = YbXZ
and AbCB = XbZY
When we describe congruent triangles, we label the vertices that are in corresponding positions
in the same order. For instance, in the previous example, we write 4ABC »= 4XYZ, not
4ABC »= 4YZX.
EXERCISE 17C
1 State whether these pairs of triangles are congruent, giving reasons for your answers:
a b c
d e f
Example 11
�
A
B
C �
X
Z
Y
�
A
B
C �
X
Z
Y
�
�
�
�
»
= means ‘is
congruent to’
� �
�
�
�
�
�
�
��
��
30°30°
60°
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Y:\HAESE\AUS_08\AUS08_17\369AUS08_17.cdr Tuesday, 16 August 2011 1:05:21 PM BEN
370 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
2 State whether these pairs of triangles are congruent, giving reasons for your answers:
a b c
d e f
3 Which of the following triangles is congruent to
the one alongside?
A B C D
4 Which of these triangles are congruent to each other?
A B C
D E F
50°
100°50°
30°
80°8 m
5 m
80°8 m
5 m
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5 m
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5 m
8 m
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80°
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50°
17 cm
9 cm
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Y:\HAESE\AUS_08\AUS08_17\370AUS08_17.cdr Tuesday, 16 August 2011 1:06:55 PM BEN
Discussion
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 371
5
i
ii
a b
c d
e f
g h
We have seen that if two triangles have equal
corresponding sides, then they are congruent.
Is the same true for quadrilaterals? Can we say
that the quadrilaterals alongside are congruent?
A
B
C
Q
P
R
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K L
X
Z
Y
D
F
P Q
RE
A B
E D
C
P
T
R
S
Q
�
T
R
S
Z
X
Y
�
For each of the following pairs of triangles, which are not drawn to scale:
Determine whether the triangles are congruent.
If the triangles are congruent, what else can we deduce about them?
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�
A
B C
D
F
E
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F
X
Y
W
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Y:\HAESE\AUS_08\AUS08_17\371AUS08_17.cdr Tuesday, 16 August 2011 1:10:58 PM BEN
372 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
In Chapter 8, we studied the properties of isosceles triangles and special quadrilaterals. We can
use congruence to prove many of these properties.
Self Tutor
Consider the isosceles triangle ABC.
M is the midpoint of [BC].
a Use congruence to show that BbAM = CbAM.
b What property of isosceles triangles has been
proven?
a In triangles ABM and ACM: ² AB = AC f4ABC is isoscelesg
² BM = CM fM is the midpoint of [BC]g
² [AM] is common to both triangles.
) 4ABM »= 4ACM fSSSg
Equating corresponding angles, BbAM = CbAM.
b In an isosceles triangle, the line joining the apex to the
midpoint of the base bisects the vertical angle.
EXERCISE 17D
1 Consider the parallelogram ABCD.
a Copy and complete:
In triangles ABD and CDB:
² AbDB = ...... falternate anglesg
² AbBD = ...... falternate anglesg
² [BD] is common to both triangles
) 4ABD »= 4CDB f......g
Equating corresponding angles, DbAB = ......
b What property of parallelograms has been proven in a?
2 Consider the kite PQRS.
a Use congruence to show that QbPR = SbPR and
QbRP = SbRP.
b What property of kites has been proven?
PROOF USING CONGRUENCED
Example 12
A
B CM
A B
CD
P
Q
R
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Y:\HAESE\AUS_08\AUS08_17\372AUS08_17.cdr Tuesday, 16 August 2011 1:11:29 PM BEN
Review set 17
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 373
3 Consider the square ABCD.
a Show that 4ABC »= 4DAB.
b Hence, show that AC = DB.
c What property of squares has been proven?
4 Consider the rhombus WXYZ.
a Show that 4WXY »= 4YZW.
b Hence, show that XbYW = Z bWY.
c Explain why [XY] is parallel to [WZ].
d Likewise, show that [XW] is parallel to [YZ].
e What property of rhombuses has been proven?
5 The diagonals of rhombus PQRS meet at M.
a Show that 4PSQ »= 4RSQ.
b Hence, show that PbSQ = RbSQ.
c What property of rhombuses has been proven?
d Explain why 4PSM »= 4RSM.
e Hence:
i show that PM = RM
ii find the sizes of SbMP and SbMR.
f Use e to show that 4SMP »= 4QMR, and therefore SM = QM.
g What property of rhombuses has been proven in e and f?
6 Use congruence to show that:
a the opposite sides of a parallelogram are equal in length
b the base angles of an isosceles triangle are equal
c the diagonals of a kite intersect at right angles.
1 Translate the given figures in the direction indicated:
a
3 units right
b
2 units left and 2 units down
A B
CD
X Y
ZW
Q R
SP
M
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Y:\HAESE\AUS_08\AUS08_17\373AUS08_17.cdr Monday, 22 August 2011 10:50:43 AM BEN
374 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
2 Describe the following transformations:
a b
3 Rotate each figure about O through the angle indicated:
a
180±
b
270± anticlockwise
4 Reflect each figure in the axis indicated:
a
x-axis
b
y-axis
5 a State the vertex coordinates of the
quadrilateral PQRS alongside.
b Rotate the quadrilateral 90± anticlockwise
about the origin O.
c State the vertex coordinates of the image.
6 a Which two of the figures alongside are
translations of each other?
b Which two of the figures are reflections
of each other?
c Which of the axes is the mirror line for
this reflection?
A
B C
D
A
B C
D0
0
00 AB
C
D
0
E F
BA
F E
C
D
0
0
0
00
OO
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xQ
R
SP
3
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y
x
A
C
B
D
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Y:\HAESE\AUS_08\AUS08_17\374AUS08_17.cdr Tuesday, 16 August 2011 1:13:42 PM BEN
Practice test 17A Multiple Choice
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 375
7 Show that A is congruent to B by transforming A to B in a single transformation:
a b
8 Consider the kite ABCD.
Use congruence to show that AbBC = AbDC.
What property of kites has been proven?
9
i Determine whether the triangles are congruent.
ii If the triangles are congruent, what can be deduced from the congruence?
a b
10 Show that A is congruent to B by transforming A to B in a combination of transformations:
a b
11 Use congruence to prove that, in an isosceles triangle, the line joining the apex to the
midpoint of the base meets the base at right angles.
Click on the link to obtain a printable version of this test.
A
B
C
D
PRINTABLE
TEST
y
x
B Ay
x
A
B
A
B
C X
Y
Z
y
x
B
A
y
x
A
B
For each of the following pairs of triangles, not drawn to scale:
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FE
D
S U
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Y:\HAESE\AUS_08\AUS08_17\375AUS08_17.cdr Tuesday, 16 August 2011 1:16:51 PM BEN
Practice test 17B Short response
376 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
1 a
Describe the translation from R to S.
b
Copy the given figure and translate it
5 units left and 2 units up.
2 a Translate the quadrilateral ABCD
2 units right and 3 units down.
b State the vertex coordinates of the
image.
3 Copy the figure and reflect it in the
mirror line shown.
4 Show that P is congruent
to Q by a single
transformation of P to Q.
a b
5 A triangle has vertices A(2, 1), B(4, 3), and C(3, 0).
a Plot triangle ABC on a Cartesian plane.
b Reflect 4ABC in the x-axis, and state the vertex coordinates of the image.
c Reflect 4ABC in the y-axis, and state the vertex coordinates of the image.
6 State whether these pairs of triangles are congruent, giving reasons for your answers.
a b
R
S
y
x
A
DC
B
y
x
P Q
y
x
P
Q
m
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Y:\HAESE\AUS_08\AUS08_18 - new 17\376AUS08_17.cdr Monday, 1 August 2011 12:09:13 PM BEN
Practice test 17C Extended response
CONGRUENCE AND TRANSFORMATIONS (Chapter 17) 377
7 a Which of the figures alongside is a
rotation of A about the origin?
b Determine the angle of rotation from
figure A to this figure.
8 State whether each pair of figures is congruent, giving reasons for your answer:
a b c
9 Show that A is congruent to B by transforming A to B in a combination of transformations:
a b
10 State whether these pairs of triangles are congruent, giving reasons for your answers:
a b c
1 Triangle T has coordinates (¡2, 1), (1, 3), and (2, 1).
a Plot the triangle T on a Cartesian plane.
b Translate T 4 units right and 2 units up. State the vertex coordinates of the image
triangle T0.
c Translate T0 1 unit left and 5 units down. State the vertex coordinates of the image
triangle T00.
d Describe a single transformation from T to T00.
y
x
A
D
C
B
y
x
A
B
y
x
A
B
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Y:\HAESE\AUS_08\AUS08_17\377AUS08_17.cdr Tuesday, 16 August 2011 1:18:26 PM BEN
378 CONGRUENCE AND TRANSFORMATIONS (Chapter 17)
2 A square piece of paper is divided into four
triangles as shown.
a Show that triangles A and D are congruent.
b Hence, show that triangles B and C are
congruent.
c Find the area of each triangle.
3 a Which of the figures alongside is a translation
of A?
b Which figure is a reflection of A?
c Which figure is a rotation of A?
d Describe the transformation from C to D.
e Are all of the figures congruent? Explain
your answer.
4 Consider the kite ABCD alongside.
a Use congruence to show that BbAC = DbAC.
b Hence, show that 4ABX »= 4ADX.
c Show that [BX] and [DX] have the same length.
d What property of kites has been proven in c?
5 Consider the isosceles triangle ABC alongside.
Each angle of the triangle is trisected, or
divided into 3 equal parts. The angle trisectors
meet at D, E, and F as shown.
a Show that BF = CF.
b Show that 4ABD »= 4ACE, and hence
BD = CE.
c Show that 4BDF »= 4CEF.
d Hence, show that triangle DEF is also
isosceles.
B
D
A CX
B C
A
F
ED
y
x
A B
D C
A
C
B
D
5 cm
15 cm
P U
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Y:\HAESE\AUS_08\AUS08_17\378AUS08_17.cdr Tuesday, 16 August 2011 1:22:43 PM BEN