Geometry slides Year 9 NZ
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Transcript of Geometry slides Year 9 NZ
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Angles within a right angle add to 90°
• They are called complementary angles
60°
a=? a= 30°
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Isosceles Triangles
• The angles opposite the equal sides are equal angles.
73° b= ?
b= 73°
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PolygonsClosed figures made up of straight
sides.
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Examples of Polygons
• Triangle• Quadrilateral• Pentagon• Hexagon• Octagon• Decagon
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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.
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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°.
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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
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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.
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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
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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.
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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
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Navigation and Bearings
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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
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Bearings
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Starter
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Paper Cutting
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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.
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E
D
A B
F
G
A'
C
E’
F’
B’
D’ C’
G’
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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
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m
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Where would the mirror line go??
m
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Invariant Points
• If a point is already on the mirror line, it stays where it is when reflected. These points are called invariant points.
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m
m
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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
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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.
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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.
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D
A
C
B
D’
A’
B’
C’
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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
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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.
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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
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Drawing Rotations
A
B C
D B’
D’ C’
¼ turn clockwise =
90º clockwise
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Drawing Rotations
A
B
C
B’
¼ Turn anti-clockwise = 90º Anti-clockwise
C’
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By what angle is this flag rotated about point C ?
180º
Remember: Rotation is always measured in the anti clockwise direction!
C
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By what angle is this flag rotated about point C ?
90º
C
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By what angle is this flag rotated about point C ?
270º
C
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Questions• Ex 26.07 Page 411 Qn 1, 2, 3, 7 and 8.
• Ex 26.08 Page 412 Qn 1, 3, 4 and 6.
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Answers 26.07
A
D C
B
B’C’
D’ A’
A
A’
B
C
B’
C’
1
2
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A
D
BB’
C’ D’
A’
3
C
7
D’
A
D
B’C’
A’
B
C
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ss
8
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Ex. 26.08
1. a) P b) R c) QS
3. 180°
4. 0° or 360°
8. a) R b) Q c) CB
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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
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• 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
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m
A
C
B
D
D’
A’ B’
C’D’’
B’’C’’
A’’C’’’ B’’’
D’’’ A’’’
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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.
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Order of rotational symmetry= Order of rotational symmetry=
34
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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
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Total order of symmetry
• The number of axes of symmetry plus the order of rotational symmetry.
2+2=4
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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.
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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.
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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
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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).
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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
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Translations
Each point moves the same distance in the same direction
There are no invariant points in a translation(every point moves)
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← movement in the x direction (right and left)
← movement in the y direction (up and down)( )yx
+
+
-
-
• Vectors describe movement
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Vectors
• Vectors describe movement
Each vertex of shape EFGH moves along the vector
( )-3
-6
To become the translated shape E’F’G’H’
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• Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`.
( )- 4
- 2
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Vectors
-5 +4
+2 -6
-6 -2
+3 +4
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Object
Image
6
1
Object
Image
-12
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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.
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Scale factor of 2
Scale factor of 1/2
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Scale Factor• The scale factor tells us how much the lengths
of an object are multiplied by to get the lengths of the image.