Chapters 5, 6, 9 : Measurement and Geometry I and II

1
Chapters 5, 6, 9 Measurement and Geometry I and
II
2
A conjecture

• A conjecture is a mathematical statement that
  appears likely to be true, based on evidence (or
  observation) but has not been proven.
• Conjecture is used constantly in Geometry and
  Geometric Proofs.

3
Intersecting Lines and Line Segments

• When two lines intersect, four angles are
  produced.
• Opposite angles are equal in measure.
• Two angles that add to 180 are supplementary.
• Two angles that add to 90 are complementary.

4
Perpendicular Line Segments

• Perpendicular line segments are line segments
  that intersect at a 90 angle upward or downward.
• Perpendicular lines have slopes that are negative
  reciprocals of each other (see example on
  chalkboard)

5
Parallel Line Segments

• Parallel line segments are line segments that
  never intersect.
• Parallel lines have identical slopes (Chapter 2)
  but different start points (y-intercepts)

6
Parallel Lines Theorems

• When a line intersects parallel lines, three
  angle theorems are formed (see page 259)
• Alternate Interior Angles theorem (Z pattern)
• Corresponding Angles theorem (F pattern)
• Supplementary Interior Angles theorem (C pattern)

7
Common prefixes

• Prefixes are always attached to the beginning of
  a word and mean a specific thing.

• Tri 3
• Tetra 4
• Penta 5
• Hexa 6
• Hepta 7
• Octa 8
• Nona 9
• Deca 10
• etc

8
A polygon

• A polygon has all sides congruent and all angles
  congruent.
• Polygons can be both regular and irregular.
• Regular polygons have both reflective and
  rotational symmetry. (Major difference between
  regular and irregular polygons)

9
Examples of common regular polygons

• Regular trigon (equilateral triangle)
• Regular tetrahedron (square)
• Regular pentagon
• Regular hexagon
• Regular octagon

10
The prefixes of the metric system

• Here are the frequently used prefixes of the
  metric system

• Kilo- (k) 1000
• Hecto- (h) 100
• Deca- (da) 10
• Base- 1
• Deci- (d) 1/10 0.1
• Centi- (c) 1/100 0.01
• Milli- (m) 1/1000 0.001

11
Converting measurements into different metric
units

• To convert metric units, you must use the metric
  staircase.

12
How to use the metric staircase 1

• When you go down the staircase, you are
  converting from a larger unit to a smaller unit.
  
• So, you multiply the given number by 10number of
  steps

13
An example of conversion 1

• To convert 6 km to meters
• 6 km (6 x 103) m
• 6 km (6 x 1000) m
• 6 km 6000 m

14
How to use the metric staircase 2

• When you go up the staircase, you are converting
  from a smaller unit to a larger unit.
• So, you divide the given number by 10number of
  steps

15
An example of conversion 2

• To convert 1200 mL to liters
• 1200 mL (1200 103) L
• 1200 mL (1200 1000) L
• 1200 mL 1.2 L

16
The perimeter

• The perimeter is the total distance around a
  figures outside.
• The symbol of perimeter is P.
• The perimeter is a one-dimensional quantity
  measured in linear units (an exponent of 1) ,
  such as millimeters, centimeters, meter or
  kilometers.

17
Area

• Area is the measure of the size of the region it
  encloses.
• The symbol of area is A.
• Area is a two-dimensional quantity measured in
  square units (an exponent of 2) such as
  centimeters squared, meters squared or kilometers
  squared.

18
Area of a rectangle

• To calculate the area of a rectangle
• Arectangle length x width

19
Area of a triangle

• To calculate the area of a triangle
• Atriangle ½ x base x height

20
A composite figure

• A composite figure is a figure that is made of 2
  or more common shapes or figures.
• For example, you can break up a pentagon (a
  composite figure) into a rectangle and a
  triangle.

21
What is a circle?

• A circle is a 2-dimensional geometric shape
  consisting of all the points in a plane that are
  a constant distance from a fixed point.
• The constant distance is called the radius of the
  circle.
• The fixed point is called the centre of the
  circle.
• There are 360 in a complete rotation around a
  circle.

22
What is pi?

• Pi is an irrational number that states the ratio
  of the circumference of a circle to its diameter.
  
• The symbol for pi is ?
• Its value is 3.1412 (it is a non repeating
  decimal value)
• To make life easier, we will assume that the
  value of pi is 3.

23
The circumference of a circle

• The circumference of a circle is the distance
  around the circle.
• So, the circumference is the perimeter of a
  circle.
• The symbol of the circumference is C.

24
How to calculate the circumference

• To calculate the circumference of a circle
• C (2)(?)(r) or C(?)(d)
• r is the radius of the circle
• d is the diameter of the circle.

25
How to calculate the area of a circle

• To calculate the area of a circle
• A (?)(r2)

26
Geometry vocabulary terms

• Congruent means the same size and the same shape.
  
• Parallel means in the same plane but no
  intersection.
• Un net can help visualize the faces of a 3-D
  figure (see page 221)
• Collinear means that all points are in the same
  straight line.

27
Prisms and cylinders

• Prisms and cylinders have 2 faces that are
  congruent and parallel.

28
Examples of prisms and cylinders

• There are 3 common examples

• a rectangular prism
• a cylinder
• a triangular prism

29
Surface area of prisms and cylinders

• The surface area of a prism is equal to the sum
  of all its outer faces.
• The surface area of a cylinder is equal to the
  sum of all its outer faces.

30
3-D composite figures

• A composite 3-D figure/shape is made up of 2 or
  more 3-D shapes/figures.

31
Surface area of 3-D composite figures

• To determine the surface area of three
  dimensional figure is the total outer area of all
  its faces.
• So, the surface area is equal to the sum of all
  its faces (add them all together)

32
The volume of prisms and cylinders

• The volume of a solid is the amount of space it
  occupies.
• The symbol of volume is V.
• The volume is a three-dimensional quantity,
  measured in cubic units (an exponent of 3), such
  as millimeters cubed, centimeters cubed, and
  meters cubed.

33
The capacity of prisms and cylinders

• The capacity is the greatest volume that a
  container can hold.
• The capacity is measured in liters or milliliters.

34
How to calculate the volume of a prism

• To calculate the volume of a prism
• Vprism area of the prisms base x prisms
  height
• Vprism Abase x h

35
How to calculate the volume of a cylinder

• To calculate the volume of a cylinder
• Vcylindre ?r2 x h

36
How to calculate the volume of 3-D composite
figures

• You can find the volume of a 3-D composite figure
  by adding the volumes of the figures that make up
  the 3-D shape.

37
The volume of 3-D figures

• The volume is the space that an object occupies,
  expressed in cubic units.
• A polygon is a two-dimensional closed figure
  whose sides are line segments.
• A polyhedron is a three-dimensional figure with
  faces that are polygons. The plural is
  polyhedra.

38
3-D figures

• We are going to calculate the volume of three 3-D
  figures

1. A cone
2. A pyramid
3. A sphere

39
A cone

• A cone is a 3-D object with a circular base and a
  curved surface.

40
How to calculate the volume of a cone

• To calculate the le volume of a cone
• Vcône 1/3 x (the volume of cylinder)
• Vcône 1/3 x ?r2 x h

41
A pyramid

• A pyramid is a polyhedron with one base and the
  same number of triangular faces as there are
  sides on the base.
• Like prisms, pyramids are named according to
  their base shape.

42
How to calculate the volume of a pyramid

• To calculate the volume of a pyramid
• Vpyramide 1/3 x (the volume of prism)
• Vpyramide 1/3 x Abase x h

43
A sphere

• A sphere is a round ball-shaped object.
• All points on the surface are the same distance
  from a fixed point called the centre.

44
How to calculate the volume of a sphere

• To calculate the volume of a sphere
• Volume of a sphere 4/3 x ?r3

45
Surface area of 3-D figures

• Surface area is the sum of all the areas of the
  exposed faces of a 3-D figure.
• The symbol for surface area is At

46
How to calculate the surface area of a cylinder

• To calculate the surface area of a cylinder
• At 2?r2 2?rh

47
How to calculate the surface area of a cone

• To calculate the surface area of a cone
• It is the sum of the base area and the lateral
  area.
• At ?r2 ?ro

48
The slant height

• The length of the slant height uses the symbol s
• The slant height is calculated by using the
  Pythagorean relationship.

49
How to calculate the surface area of a sphere

• To calculate the surface area of a sphere
• At 4?r2

50
A cube

• A cube is the product of three equal factors.
• Each factor is considered the cube root of this
  particular cube/product.
• For example, the cube root of 8 is 2 because 23
  2 x 2 x 2 8

51
Unique Triangles

• A unique triangle is a triangle that does not
  have an equivalent. (one-of-a-kind)

52
How to create a unique triangle

• These conditions are needed to create a unique
  triangle
• The SSS case means that all three sides are
  given.
• The SAS case means that the measures of two sides
  and the angle between the two sides are given.
• The ASA case means that the two angles and the
  side contained between the two angles are given.
• The AAS case means that the two angles and a
  non-contained side are given.

53
Congruence

• The symbol for congruence, , is read is
  congruent to.
• If 2 geometric figures are congruent, they have
  the same shape and size.

54
How to determine 2 Congruent Triangles

• To determine 2 congruent triangles, we must check
  a set of minimum sufficient conditions
• Measure the lengths of 1 pair of corresponding
  sides and 2 pairs of corresponding angles and
  find them equal.
• Measure the lengths of 2 pairs of corresponding
  sides and the angles included by these sides and
  find them equal.
• Measure the lengths of 3 pairs of corresponding
  sides and find them equal.

55
Similar figures

• The symbol, , means is similar to
• Two figures (polygons) are similar when their
  corresponding angles have the same measure and
  their corresponding sides are in proportion.

56
How to determine 2 Similar Triangles

• To determine 2 similar triangles, we must check a
  set of minimum sufficient conditions
• 2 pairs of corresponding angles have the same
  measure.
• The ratios of 3 pairs of corresponding sides are
  equal (i.e. these 3 pairs are proportional)
• 2 pairs of corresponding sides are proportional
  and the corresponding included angles are equal.
  

57
Transformations

• A transformation is a mapping of one geometrical
  figure to another according to some rule.
• A transformation changes a figures pre-image to
  an image.

58
Pre-image vs. Image

• A pre-image is the original line or figure before
  a transformation.
• An image is the resulting line or figure after a
  transformation.
• See page 5 of Math 9 booklet to see the
  difference in notation between these 2 terms.

59
The types of transformations

• There are 4 types of transformations

• Translations
• Reflections
• Rotations
• Dilatations.

60
A translation

• A translation is a slide. It is represented by a
  translation arrow.

61
A reflection

• A reflection is a flip. It is represented by a
  reflection line m (a double arrowed line)

62
A rotation

• A rotation is a turn. It is represented by a
  curved arrow either in a clockwise or counter
  clockwise direction.

63
A dilatation

• A dilatation is an enlargement or reduction.
  Dilatations always need a dilatation centre and a
  scaling factor.
• A scale factor is a ratio or number that
  represents the amount by which a figure is
  enlarged or reduced
• (image measurement) (pre-image measurement)

64
Transformations on a Cartesian Grid

• A map associates each point of a geometric shape
  with a corresponding point in another geometric
  shape on a Cartesian Grid.
• A map shows how a transformation changes a
  pre-image to an image.

65
An example of a map

• (2,3) ? (4,7) means that the point (2,3) maps
  onto point (4,7).
• This implies that there is a relationship between
  the 2 points.
• (2,3) and (4,7) are called corresponding points.

66
Mapping Rule

• The relationship between 2 corresponding points,
  expressed as algebraic expressions, is called a
  mapping rule.
• For example (2,3) ? (4,7) has a mapping rule
  (x,y) ? (x2, y4)

67
Properties of Transformations

• The properties of translations, reflections and
  180 rotations were discussed in Grade 8.
• These properties are summarized on the worksheet
  (GS BLM 6.2 Properties of Transformations Table)

68
Minimum Sufficient Conditions for Transformations

• To be certain that 2 shapes have undergone a
  specific transformation, one must provide a
  minimum sufficient condition (information).

69
The Minimum Sufficient Condition for a Translation

• The line segments joining corresponding points
  are congruent, parallel and in the same
  direction.

70
Minimum Sufficient Condition for a Reflection

• The line segments joining corresponding points
  have a common perpendicular bisector.

71
A perpendicular bisector

• A perpendicular bisector is a line drawn
  perpendicular (at a 90 angle) to a line segment
  dividing it into 2 equal parts.
• The perpendicular bisector always intersects with
  the midpoint of the original line segment.

72
Minimum Sufficient Condition for a 180 Rotation

• The line segments joining corresponding points
  intersect at their midpoints.

73
Regular polyhedron (Grade 7)

• A regular polyhedron is a 3-D figure with faces
  that are polygons.
• Polyhedrons plural is polyhedra.

74
Platonic solids

• The Platonic solids are the 5 regular polyhedra
  named after the Greek Mathematician Plato.

75
The 5 Platonic solids

• The cube
• The regular tetrahedron
• The regular octahedron
• The regular dodecahedron
• The regular icosahedron
• See Page 39 of Math 9 booklet

76
The 3 characteristics of regular polyhedra
(Platonic solids)

1. All faces are 1 type of regular polygon.
2. All faces are congruent.
3. All vertices are the same (i.e. they have vertex
   regularity)

77
What is vertex regularity?

• When all vertices in a polyhedron are the same,
  you have vertex regularity, which can be
  described using notation.
• For example, the notation 5,5,5 represents the
  vertex regularity of a regular dodecahedron
  because there are 3 regular 5-sided polygons at
  every vertex.

78
Circle Geometry

• In this section of circle geometry, we will be
  introduced to these new terms

• Central angles
• Inscribed angles
• Tangent of a circle
• Circumscribed angle

79
Central angle

• A central angle is an angle formed by 2 radii of
  a circle. (page 42)

80
Inscribed angle

• An inscribed angle is an angle that has its
  vertex on a circle and is subtended by an arc of
  the circle. (page 42)
• What does subtended mean geometrically?

81
Tangent of a circle

• A tangent of a circle is a line that touches a
  circle at only 1 point, the point of tangency.
  (page 43)

82
Circumscribed angle

• A circumscribed angle is an angle with both arms
  tangent to a circle. (page 44)

83
A polygon

• A polygon has all sides congruent and all angles
  congruent.
• Polygons can be both regular and irregular.
• Regular polygons have both reflective and
  rotational symmetry. (Major difference between
  regular and irregular polygons)

84
Regular polyhedron

• A regular polyhedron is a 3-D figure with faces
  that are polygons.
• Polyhedrons plural is polyhedra.

85
Polyhedra with regular polygonal faces

• In grade 9 Geometry, there are several types of
  polyhedra

• The 5 Platonic solids
• A uniform prism
• An antiprism
• A deltahedron
• A dipyramid
• The Archimedean solids

86
The 5 Platonic solids

• The cube
• The regular tetrahedron
• The regular octahedron
• The regular dodecahedron
• The regular icosahedron
• See Page 39 of Math 9 booklet

87
Uniform prism

• A uniform prism is a prism having only regular
  polygonal faces. (page 50)

88
Antiprism

• An antiprism is a polyhedron formed by 2
  parallel, congruent bases and triangles.
• Each triangular face is adjacent (next to) 1 of
  the congruent bases.
• Page 51

89
Deltahedron

• A deltahedron is a polyhedron that has only
  equilateral triangle faces.
• The deltahedron is named after the Greek symbol
  delta (?)
• The plural is deltahedra.
• Page 51

90
Dipyramid

• A dipyramid is a polyhedron with all triangle
  faces formed by placing 2 pyramids base to base.
  
• Page 52

91
Archimedean solids

• The Archimedean solids are the 13 different
  semi-regular polyhedra.
• The Archimedean solids have vertex regularity and
  symmetry (reflective and rotational)

92
13 Archimedean solids (page 53)

• Cuboctahedron
• Great rhombicosidodecahedron
• Great rhombicuboctahedron
• Icosidodecahedron
• Small rhombicosidodecahedron
• Small rhombicuboctahedron
• Snub cube

93
13 Archimedean solids (page 53) continued

• Snub dodecahedron
• Truncated dodecahedron
• Truncated icosahedron
• Truncated octahedron
• Truncated tetrahedron
• Truncated cube

94
What is a vertex?

• A vertex is a point at which 2 or more edges of a
  figure meet.
• The plural is vertices.

95
Vertex configuration

• Vertex configuration is the arrangement of
  regular polygons at the vertices of a polyhedron.
  (page 50)
• Vertex configuration notation refers to the types
  of regular polygons around a vertex.
• For example, the notation 3,4,5,4 means that a
  vertex has an equilateral triangle, a square, a
  regular pentagon and a square around it in that
  order.

96
Plane of symmetry

• A plane of symmetry is a plane dividing a
  polyhedron into 2 congruent halves that are
  reflective images across the plane.
• Page 53

97
Axis of symmetry

• An axis of symmetry is a line about which a
  polyhedron coincides with itself as it rotates.
• The number of times a polyhedron coincides with
  itself in 1 complete rotation is its order of
  rotational symmetry.

98
The properties of regular polyhedra

1. All faces are regular polygons.
2. All faces are the same type of congruent polygon.
3. The same number of faces meet at each vertex.
4. Regular polyhedra have several axis of symmetry
   (rotational symmetry)
5. Regular polyhedra have several planes of symmetry
   (reflective symmetry)

99
The difference between semi-regular and regular
polyhedra

• Regular polyhedra Platonic solids, etc.
• Semi-regular polyhedra Archimedean solids
• All faces of a semi-regular polyhedron are not
  the same type of regular polygon.