geometry slides year 9 nz

Angles within a right angle add to 90°

• They are called complementary angles

60°

a=? a= 30°

Isosceles Triangles

• The angles opposite the equal sides are equal angles.

73° b= ?

b= 73°

PolygonsClosed figures made up of straight

sides.

Examples of Polygons

• Triangle• Quadrilateral• Pentagon• Hexagon• Octagon• Decagon

Angles of Polygons• Interior angles- are angles

between the sides of thepolygon on the inside.

• Exterior angles- are angles found by extending the sides of the polygon.

d

Exterior Angles

• Measure the exterior angles of your polygons.• Add the exterior angles of each shape together.• What do they add to?

Shape Total Degrees of Exterior Angles

Triangle

Quadrilateral

Pentagon

Hexagon

360

360

360

360

The sum of the exterior angles of a polygon is 360°.

Alternate angles When two parallel lines are cut by a third line, then angles in alternate

positions equal in size.

Co-interior angles When two parallel lines are cut by a third line, co-interior angles are

supplementary.

Angles at a point. The sum of the sizes of the angles at a point is 360

Adjacent angles on a straight line

The sum of the sizes of the angles on a line is 180 degrease

Adjacent angles in a right angle

The sum of the size's of the angles in around different points but the same angle

Vertically opposite angles

Vertically opposite angles are equal in size.

Corresponding angles

When two parallel lines are cut by a third line, then angles in corresponding positions are equal in size.

ttrsgf

Interior Angles of a PolygonNo. of sides of polygon

3 4 5 You do 6, 8, 10

Drawing

Number of triangles

1 2 3

Degrees in a triangle sum to 180°If there are 180 degrees in a triangle how many degrees must thereBe in a quadrilateral which is split into 2 triangles.

The rule (n 2) × 180° n is the number of sidesof the polygon

Regular Polygons

• A polygon is called regular if all its sides are the same length and all its angles are the same size.

e.g. equilateral triangle, a square or a regular pentagon.

Exterior Angles Number of sides

Sum of exterior angles

Each exterior angle

Equilateral triangleSquarePentagonHexagonOctagonDecagon

Interior Angles Number of sides

Sum of exterior angles

Each exterior angle

Equilateral triangle SquarePentagonHexagonOctagonDecagon

34568

10

3

456810

360360360360360

360

12090726045

36

180

36054072010801440

60

90108120135144

Navigation and Bearings

Bearings• Bearings are angles which are measured clockwise

from north. They are always written using 3 digits. • The bearings start at 000 facing north and finish at

360 facing north.

Bearings

045

120

180

270

Bearings

Starter

Paper Cutting

Translation• A translation is a movement in which each

point moves in the same direction by the same distance.

• To translate an object all you need to know is the image of one point. Every other point moves in the same distance in the same direction.

E

D

A B

F

G

A'

C

E’

F’

B’

D’ C’

G’

Reflection

• In a reflection, and object and its image are on opposite sides of a line of symmetry.

• This line is often called a mirror line.

m

m

Where would the mirror line go??

m

Invariant Points

• If a point is already on the mirror line, it stays where it is when reflected. These points are called invariant points.

m

m

Exercises Today

• Ex 26.03 2 of 1a, b, c or d. Page397 • 26.04 Question 2, 7 and 8. Page 401 & 402• Any three questions from 26.05. Page 404

Rotation • Rotation is a transformation where an object

is turned around a point to give its image.• Each part of the object is turned through the

same angle. • To rotate an object you need to know where

the center of rotation is and the angle of rotation.

The angle of rotation

• This can be given in degrees or as a fraction such as a quarter turn.

• The direction the object is turned can be either clockwise or anti-clockwise.

D

A

C

B

D’

A’

B’

C’

Starter• For the numbers below think of what they could

mean in the world. Get as many answers as possible.

366

4 000 000

3015

25

11

Rotation• In rotation every point rotates through a certain

angle.• The object is rotated about a fixed point called

the centre of rotation. • Rotation is always done in an anti-clockwise

direction.• A point and it’s image are always the same

distance from the centre of rotation.• The centre of rotation is the only invariant point.

What are these equivalent angles

of Rotation?• 270º Anti clockwise is _______ clockwise• 180º Anti clockwise is ______ clockwise• 340º Anti clockwise is _______ clockwise

Rotations are always specified in the anti clockwise direction

Drawing Rotations

A

B C

D B’

D’ C’

¼ turn clockwise =

90º clockwise

Drawing Rotations

A

B

C

B’

¼ Turn anti-clockwise = 90º Anti-clockwise

C’

By what angle is this flag rotated about point C ?

180º

Remember: Rotation is always measured in the anti clockwise direction!

C

By what angle is this flag rotated about point C ?

90º

C

By what angle is this flag rotated about point C ?

270º

C

Questions• Ex 26.07 Page 411 Qn 1, 2, 3, 7 and 8.

• Ex 26.08 Page 412 Qn 1, 3, 4 and 6.

Answers 26.07

A

D C

B

B’C’

D’ A’

A

A’

B

C

B’

C’

1

2

A

D

BB’

C’ D’

A’

3

C

7

D’

A

D

B’C’

A’

B

C

ss

8

Ex. 26.08

1. a) P b) R c) QS

3. 180°

4. 0° or 360°

8. a) R b) Q c) CB

Define these terms• Mirror line• Centre of rotation• Invariant

What is invariant in• Reflection

• rotation

The line equidistant from an object and its imageThe point an object is rotated aboutDoesn’t change

The mirror line

Centre of rotation

• Draw a rectangle. Have two units across and 3 going up. (Drawn below)

• Label the rectangle A, B, C and D.• Reflect the object in the line CD. Label the image A’,

B’, C’ and D’.• Rotate the image about point D’ 90° anticlockwise.

Label this image A’’, B’’, C’’ and D’’.• Translate the 2nd image by the vector . Label the

rectangle A’’’’, B’’’’, C’’’’ and D’’’’.

C

A B

D

m

A

C

B

D

D’

A’ B’

C’D’’

B’’C’’

A’’C’’’ B’’’

D’’’ A’’’

Rotational Symmetry

• A figure has rotational symmetry about a point if there is a rotation other than 360° when the figure can turn onto itself.

• Order of rotational symmetry is the number of times a figure can map onto itself.

Order of rotational symmetry= Order of rotational symmetry=

34

Line Symmetry

• A shape has line symmetry if it reflects or folds onto itself.

• The fold is called an axis of symmetry.

Line symmetry= 2

Total order of symmetry

• The number of axes of symmetry plus the order of rotational symmetry.

2+2=4

Mathematical Dance• You must prepare a dance lasting 1 minute for

tomorrow’s lesson. • You must use moves demonstrating reflection,

rotation and translation in your dance. • You must have a sheet with your dance moves

recorded using mathematical language.• One person in the group must call out the

dance moves during the performance whilst dancing with their group.

Mathematical Dance

• You have 10 minutes today and 5 minutes tomorrow to prepare your masterpiece.

• You can bring sensible music and consumes for your dance tomorrow.

Groups• Group 1- Claudia, Martine, Amiee and Emily • Group2- Olivia, Bailey, Rachael and Elle• Group3- Emma H, Erin, Michal and Charlotte• Group4- Abby, Mia, Ashleigh and Brittany• Group5- Payton, Shannon, Tayla and Susan

0 5 6 7 81-8 -7 -6 -5 -4 -3 -2 -1 42 3

0 5 6 7 81-8 -7 -6 -5 -4 -3 -2 -1 42 3

1. Move the red dot by the following values and state where it now lies.(-1), (4), (7), (-4) and (11).

1. Move the green dot by the following values and state where it now lies.(-6), (3), (-9), (-4) and (5).

Groups• Group One- Kelly, Ruby, Bella, Chrisanna and Bianca• Group Two- Hannah B, Grace, Shanice, Grace and

Georgia R• Group Three- Hannah C, Remy, Olivia, Kendyl and

Sarah• Group Four- Kelsey, Shaquille, Kiriana, Cadyne and

Claudia• Group Five- Lauran, Ashlee, Sophie, Georgia W and

Emily S • Group Six- Esther, Emily M, Jemma, Amelia and Julia

Translations

Each point moves the same distance in the same direction

There are no invariant points in a translation(every point moves)

← movement in the x direction (right and left)

← movement in the y direction (up and down)( )yx

+

+

-

-

• Vectors describe movement

Vectors

• Vectors describe movement

Each vertex of shape EFGH moves along the vector

( )-3

-6

To become the translated shape E’F’G’H’

• Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`.

( )- 4

- 2

Vectors

-5 +4

+2 -6

-6 -2

+3 +4

Object

Image

6

1

Object

Image

-12

Enlargement• When an object is enlarged its size changes.• A scale factor tells us how much larger a

shape is after it has been enlarged.

Scale factor of 2

Scale factor of 1/2

Scale Factor• The scale factor tells us how much the lengths

of an object are multiplied by to get the lengths of the image.