Chapter · than mine”. Jane used a ruler to measure the sides of each slice. “See, my slice has...

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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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

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B

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D

B

A

D C

B

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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

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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

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B

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CD

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CD A0 B0

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Investigation 1 Reflections


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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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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m

m m

mm

m

y

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y

xA 0A

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A

m

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A


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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

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x

A

A0

y

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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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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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Discussion


3 a State the vertex coordinates of triangle PQR.

b Rotate the triangle 90± anticlockwise about the

origin O.


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.


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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Activity Creating congruent figures


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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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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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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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.

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y

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Investigation 2 Congruent triangles


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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Investigation 3 Constructing triangles


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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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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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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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

70°8 m

5 m

80°

5 m

8 m

8 m

5 m

80°

15 cm

50°

17 cm

9 cm

15 cm40°

9 cm

50°

40°

15 cm

17 cm�

17 m

15 m9 m

�


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Discussion


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

��

� � J

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?

�

�

A

B C

D

F

E

�

�D E

F

X

Y

W


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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

S


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Review set 17


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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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.


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

y

x

y

xQ

R

SP

3

¡3

¡4 4

y

x

A

C

B

D

y

x


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Practice test 17A Multiple Choice


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


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:

�

��

�

FE

D

S U

T


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Practice test 17B Short response


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


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


a b


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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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

R S

T

Q


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