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

A conjectureA conjecture

A A conjectureconjecture is a mathematical is a mathematical statement that appears statement that appears likelylikely to to be true, based on evidence (or be true, based on evidence (or observation) but has not been observation) but has not been proven.proven.

Conjecture is used constantly in Conjecture is used constantly in

Geometry and Geometric ProofsGeometry and Geometric Proofs. .

Intersecting Lines and Intersecting Lines and Line SegmentsLine Segments

When two lines intersect, When two lines intersect, four four angles are producedangles are produced. .

Opposite anglesOpposite angles are equal in are equal in measure.measure.

Two angles that add to 180° are Two angles that add to 180° are supplementarysupplementary..

Two angles that add to 90° are Two angles that add to 90° are complementarycomplementary..

Perpendicular Line Perpendicular Line SegmentsSegments Perpendicular line segmentsPerpendicular line segments are are

line segments that intersect at a line segments that intersect at a 9090° angle upward or downward.° angle upward or downward.

Perpendicular lines have slopes Perpendicular lines have slopes that are negative reciprocals of that are negative reciprocals of each other (see example on each other (see example on chalkboard) chalkboard)

Parallel Line SegmentsParallel Line Segments

Parallel line segmentsParallel line segments are line are line segments that never intersect. segments that never intersect.

Parallel lines have identical slopes Parallel lines have identical slopes (Chapter 2) but different start (Chapter 2) but different start points (y-intercepts)points (y-intercepts)

Parallel Lines Parallel Lines TheoremsTheorems When a line intersects parallel When a line intersects parallel

lines, three angle theorems are lines, three angle theorems are formed: (see page 259)formed: (see page 259)

Alternate Interior Angles theorem Alternate Interior Angles theorem (Z pattern) (Z pattern)

1.1. Corresponding Angles theorem Corresponding Angles theorem (F pattern) (F pattern)

2.2. Supplementary Interior Angles Supplementary Interior Angles theorem (C pattern)theorem (C pattern)

Common prefixesCommon prefixes

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

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

A polygonA polygon

A A polygonpolygon has all sides congruent has all sides congruent and all angles congruent. and all angles congruent.

Polygons can be both Polygons can be both regularregular and and irregularirregular..

Regular polygons have both Regular polygons have both reflective and rotational reflective and rotational symmetry. (Major difference symmetry. (Major difference between regular and irregular between regular and irregular polygons)polygons)

Examples of common Examples of common regular polygonsregular polygons Regular trigon (equilateral Regular trigon (equilateral

triangle)triangle) Regular tetrahedron (square)Regular tetrahedron (square) Regular pentagonRegular pentagon Regular hexagonRegular hexagon Regular octagonRegular octagon

The prefixes of the The prefixes of the metric systemmetric system Here are the Here are the

frequently used frequently used prefixesprefixes of the of the metric system: metric system:

Kilo- (k) = 1000Kilo- (k) = 1000 Hecto- (h) = 100Hecto- (h) = 100 Deca- (da) = 10Deca- (da) = 10 Base- = 1Base- = 1 Deci- (d)= 1/10 = Deci- (d)= 1/10 =

0.1 0.1 Centi- (c) = 1/100 = Centi- (c) = 1/100 =

0.010.01 Milli- (m) = 1/1000 Milli- (m) = 1/1000

= 0.001= 0.001

Converting Converting measurements into measurements into different metric unitsdifferent metric units To convert metric units, you must To convert metric units, you must

use the use the metric staircasemetric staircase..

How to use the metric How to use the metric staircase #1staircase #1 When you go down the staircaseWhen you go down the staircase, ,

you are converting from a larger you are converting from a larger unit to a smaller unit.unit to a smaller unit.

So, you multiply the given So, you multiply the given number by 10number by 10number of stepsnumber of steps

An example of An example of conversion #1conversion #1 To convert 6 km to metersTo convert 6 km to meters:: 6 km = (6 x 106 km = (6 x 1033) m) m 6 km = (6 x 1000) m6 km = (6 x 1000) m 6 km = 6000 m6 km = 6000 m

How to use the metric How to use the metric staircase #2staircase #2

When you go up the staircaseWhen you go up the staircase, , you are converting from a smaller you are converting from a smaller unit to a larger unit. unit to a larger unit.

So, you divide the given number So, you divide the given number by 10by 10number of stepsnumber of steps

An example of An example of conversion #2conversion #2 To convert 1200 mL to litersTo convert 1200 mL to liters:: 1200 mL = (1200 1200 mL = (1200 ÷ 10÷ 1033) L) L 1200 mL = (1200 ÷ 1000) L1200 mL = (1200 ÷ 1000) L 1200 mL = 1.2 L1200 mL = 1.2 L

The perimeterThe perimeter

The The perimeterperimeter is the total is the total distance around a figure’s distance around a figure’s outside.outside.

The The symbolsymbol of perimeter is P. of perimeter is P. The The perimeterperimeter is a one- is a one-

dimensional quantity measured dimensional quantity measured in linear units (an exponent of in linear units (an exponent of 1) , such as millimeters, 1) , such as millimeters, centimeters, meter or kilometers.centimeters, meter or kilometers.

AreaArea

AreaArea is the measure of the size of is the measure of the size of the region it encloses. the region it encloses.

The The symbolsymbol of area is A. of area is A. AreaArea is a two-dimensional is a two-dimensional

quantity measured in square units quantity measured in square units (an exponent of 2) such as (an exponent of 2) such as centimeters squared, meters centimeters squared, meters squared or kilometers squared.squared or kilometers squared.

Area of a rectangleArea of a rectangle

To calculate the To calculate the area of a area of a rectanglerectangle::

AArectanglerectangle = length x width = length x width

Area of a triangleArea of a triangle

To calculate the To calculate the area of a area of a triangletriangle::

AAtriangletriangle = = ½ x base x height½ x base x height

A composite figureA composite figure

A A composite figurecomposite figure is a figure that is a figure that is made of 2 or more common is made of 2 or more common shapes or figures. shapes or figures.

For example, you can break up a For example, you can break up a pentagon (a composite figure) pentagon (a composite figure) into a rectangle and a triangle. into a rectangle and a triangle.

What is a circle?What is a circle?

A A circlecircle is a 2-dimensional geometric is a 2-dimensional geometric shape consisting of all the points in a shape consisting of all the points in a plane that are a constant distance plane that are a constant distance from a fixed point. from a fixed point.

The constant distance is called the The constant distance is called the radiusradius of the circle. of the circle.

The fixed point is called the The fixed point is called the centrecentre of of the circle. the circle.

There are There are 360360°° in a complete rotation in a complete rotation around a circle. around a circle.

What is pi?What is pi?

PiPi is an irrational number that is an irrational number that states the ratio of the states the ratio of the circumference of a circle to its circumference of a circle to its diameter. diameter.

The symbol for pi is The symbol for pi is ∏∏ Its value is 3.1412… (it is a non Its value is 3.1412… (it is a non

repeating decimal value)repeating decimal value) To make life easier, we will assume To make life easier, we will assume

that the value of pi is 3that the value of pi is 3. .

The circumference of a The circumference of a circlecircle The The circumferencecircumference of a circle is of a circle is

the distance around the circle. the distance around the circle. So, the circumference is the So, the circumference is the

perimeter of a circleperimeter of a circle.. The The symbolsymbol of the circumference of the circumference

is C.is C.

How to calculate the How to calculate the circumferencecircumference To calculate the To calculate the circumference of circumference of

a circlea circle: : C = (2)*(C = (2)*(Π)*(r) or C=(Π)*(d)Π)*(r) or C=(Π)*(d)

r is the radius of the circler is the radius of the circle d is the diameter of the circle.d is the diameter of the circle.

How to calculate the How to calculate the area of a circlearea of a circle To calculate the area of a circle: To calculate the area of a circle: A = (Π)*(rA = (Π)*(r22))

Geometry vocabulary Geometry vocabulary terms terms CongruentCongruent means the same size means the same size

and the same shape. and the same shape. ParallelParallel means in the same plane means in the same plane

but no intersection.but no intersection. Un Un netnet can help visualize the faces can help visualize the faces

of a 3-D figure (see page 221) of a 3-D figure (see page 221) CollinearCollinear means that all points are means that all points are

in the same straight line. in the same straight line.

Prisms and cylindersPrisms and cylinders

PrismsPrisms and and cylinderscylinders have 2 faces have 2 faces that are that are congruentcongruent and and parallelparallel. .

Examples of prisms Examples of prisms and cylindersand cylinders There are 3 There are 3

common common examples:examples:

a rectangular a rectangular prismprism

a cylindera cylinder a triangular a triangular

prismprism

Surface area of prisms Surface area of prisms and cylindersand cylinders The surface area of a prismThe surface area of a prism is is

equal to the sum of all its outer equal to the sum of all its outer faces.faces.

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

3-D composite figures3-D composite figures

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

Surface area of 3-D Surface area of 3-D composite figurescomposite figures To determine the surface area of To determine the surface area of

three dimensional figure is the three dimensional figure is the total outer area of all its faces. total outer area of all its faces.

So, the surface area is equal to So, the surface area is equal to the sum of all its faces (add them the sum of all its faces (add them all together) all together)

The volume of prisms The volume of prisms and cylindersand cylinders The The volumevolume of a solid is the of a solid is the

amount of space it occupies. amount of space it occupies. The The symbolsymbol of volume is V. of volume is V. The The volumevolume is a three- is a three-

dimensional quantity, measured dimensional quantity, measured in cubic units (an exponent of 3), in cubic units (an exponent of 3), such as millimeters cubed, such as millimeters cubed, centimeters cubed, and meters centimeters cubed, and meters cubed. cubed.

The capacity of prisms The capacity of prisms and cylindersand cylinders The The capacitycapacity is the greatest is the greatest

volume that a container can hold.volume that a container can hold. The capacity is measured in The capacity is measured in litersliters

or or millilitersmilliliters..

How to calculate the How to calculate the volume of a prism:volume of a prism: To calculate the To calculate the volume of a volume of a

prismprism: : VVprismprism = area of the prism’s base = area of the prism’s base

x prism’s heightx prism’s height VVprismprism = A = Abasebase x h x h

How to calculate the How to calculate the volume of a cylinder:volume of a cylinder: To calculate the To calculate the volume of a volume of a

cylindercylinder:: VVcylindrecylindre = = ΠrΠr22 x h x h

How to calculate the How to calculate the volume of 3-D composite volume of 3-D composite figuresfigures You can find the volume of a 3-D You can find the volume of a 3-D

composite figure by adding the composite figure by adding the volumes of the figures that make volumes of the figures that make up the 3-D shape.up the 3-D shape.

The volume of 3-D The volume of 3-D figuresfigures The The volumevolume is the space that an is the space that an

object occupies, expressed in object occupies, expressed in cubic cubic unitsunits..

A A polygonpolygon is a two-dimensional is a two-dimensional closed figure whose sides are line closed figure whose sides are line segments. segments.

A A polyhedronpolyhedron is a three-dimensional is a three-dimensional figure with faces that are polygons. figure with faces that are polygons. The plural is polyhedra.The plural is polyhedra.

3-D figures3-D figures

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

1.1. A coneA cone

2.2. A pyramidA pyramid

3.3. A sphereA sphere

A coneA cone

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

How to calculate the How to calculate the volume of a conevolume of a cone To calculate the To calculate the le volume of a le volume of a

cone:cone: VVcônecône = 1/3 x (the volume of = 1/3 x (the volume of

cylinder)cylinder) VVcônecône = 1/3 x Πr = 1/3 x Πr22 x h x h

A pyramidA pyramid

A A pyramidpyramid is a polyhedron with is a polyhedron with one base and the same number one base and the same number of triangular faces as there are of triangular faces as there are sides on the base. sides on the base.

Like prisms, pyramids are named Like prisms, pyramids are named according to their base shape.according to their base shape.

How to calculate the How to calculate the volume of a pyramidvolume of a pyramid To calculate To calculate the volume of a the volume of a

pyramidpyramid:: VVpyramidepyramide= 1/3 x (the volume of = 1/3 x (the volume of

prism)prism) VVpyramidepyramide = 1/3 x A = 1/3 x Abasebase x h x h

A sphereA sphere

A A spheresphere is a round ball-shaped is a round ball-shaped object. object.

All points on the surface are the All points on the surface are the same distance from a fixed point same distance from a fixed point called called the centrethe centre. .

How to calculate the How to calculate the volume of a spherevolume of a sphere To calculate the To calculate the volume of a volume of a

spheresphere:: Volume of a sphere = 4/3 x ΠrVolume of a sphere = 4/3 x Πr33

Surface area of 3-D Surface area of 3-D figuresfigures Surface areaSurface area is the sum of all the is the sum of all the

areas of the exposed faces of a 3-areas of the exposed faces of a 3-D figure. D figure.

The symbol for surface area is AThe symbol for surface area is Att

How to calculate the How to calculate the surface area of a surface area of a cylindercylinder To calculate the To calculate the surface area of a surface area of a

cylindercylinder::

AAtt= 2= 2ΠrΠr22 + 2Πrh + 2Πrh

How to calculate the How to calculate the surface area of a conesurface area of a cone To calculate the To calculate the surface area of a surface area of a

conecone::

It is the sum of the base area and It is the sum of the base area and the lateral area.the lateral area.

AAtt = = ΠrΠr22 + Πro + Πro

The slant heightThe slant height

The length of the slant height The length of the slant height uses the uses the symbol ssymbol s

The slant height is calculated by The slant height is calculated by using the using the Pythagorean Pythagorean relationshiprelationship. .

How to calculate the How to calculate the surface area of a surface area of a spheresphere To calculate the To calculate the surface area of a surface area of a

spheresphere::

AAtt = 4Πr = 4Πr22

A cubeA cube

A A cubecube is the product of three is the product of three equal factors. equal factors.

Each factor is considered the Each factor is considered the cube rootcube root of this particular of this particular cube/product. cube/product.

For example, the cube root of 8 is For example, the cube root of 8 is 2 because 22 because 233 = 2 x 2 x 2 = 8 = 2 x 2 x 2 = 8

Unique TrianglesUnique Triangles

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

How to create a unique How to create a unique triangletriangle These conditions are needed to create a These conditions are needed to create a

unique triangle:unique triangle:1.1. The The SSS caseSSS case means that all three sides are means that all three sides are

given.given.

2.2. The The SAS caseSAS case means that the measures of two means that the measures of two sides and the angle between the two sides are sides and the angle between the two sides are given.given.

3.3. The The ASA caseASA case means that the two angles and the means that the two angles and the side contained between the two angles are given.side contained between the two angles are given.

4.4. The The AAS caseAAS case means that the two angles and a means that the two angles and a non-contained side are given.non-contained side are given.

CongruenceCongruence

The symbol for The symbol for congruencecongruence, ≈, is , ≈, is read « is congruent to. »read « is congruent to. »

If 2 geometric figures are If 2 geometric figures are congruentcongruent, they have the same , they have the same shape and size. shape and size.

How to determine 2 How to determine 2 Congruent TrianglesCongruent Triangles To determine 2 congruent trianglesTo determine 2 congruent triangles, we , we

must check a set of must check a set of minimum sufficient minimum sufficient conditionsconditions::

1.1. Measure the lengths of 1 pair of Measure the lengths of 1 pair of

corresponding sides and 2 pairs of corresponding sides and 2 pairs of corresponding angles and find them equal.corresponding angles and find them equal.

2.2. Measure the lengths of 2 pairs of Measure the lengths of 2 pairs of corresponding sides and the angles included corresponding sides and the angles included by these sides and find them equal.by these sides and find them equal.

3.3. Measure the lengths of 3 pairs of Measure the lengths of 3 pairs of corresponding sides and find them equal. corresponding sides and find them equal.

Similar figuresSimilar figures

The symbol, ~, means « is similar The symbol, ~, means « is similar to »to »

Two figures (polygons) are Two figures (polygons) are similarsimilar when their corresponding angles when their corresponding angles have the same measure and their have the same measure and their corresponding sides are in corresponding sides are in proportionproportion. .

How to determine 2 How to determine 2 Similar TrianglesSimilar Triangles To determine 2 similar trianglesTo determine 2 similar triangles, we must , we must

check a set of check a set of minimum sufficient conditionsminimum sufficient conditions::

1.1. 2 pairs of corresponding angles have the 2 pairs of corresponding angles have the same measure.same measure.

2.2. The ratios of 3 pairs of corresponding The ratios of 3 pairs of corresponding sides are equal (i.e. these 3 pairs are sides are equal (i.e. these 3 pairs are proportional)proportional)

3.3. 2 pairs of corresponding sides are 2 pairs of corresponding sides are proportional and the corresponding proportional and the corresponding included angles are equal. included angles are equal.

TransformationsTransformations

A A transformationtransformation is a mapping is a mapping of one geometrical figure to of one geometrical figure to another according to some another according to some rule.rule.

A transformation changes a A transformation changes a figure’s pre-image to an figure’s pre-image to an imageimage. .

Pre-image vs. ImagePre-image vs. Image

A A pre-imagepre-image is the original line or is the original line or figure before a transformation. figure before a transformation.

An An imageimage is the resulting line or is the resulting line or figure after a transformation. figure after a transformation.

See page 5 of Math 9 booklet to See page 5 of Math 9 booklet to see the difference in notation see the difference in notation between these 2 terms.between these 2 terms.

The types of The types of transformationstransformations There are 4 There are 4

types of types of transformationstransformations::

TranslationsTranslations ReflectionsReflections RotationsRotations Dilatations.Dilatations.

A translationA translation

A A translationtranslation is a slide. It is is a slide. It is represented by a represented by a translation translation arrowarrow. .

A reflectionA reflection

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

A rotationA rotation

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

A dilatationA dilatation

A A dilatationdilatation is an enlargement or is an enlargement or reduction. Dilatations always need a reduction. Dilatations always need a dilatation centredilatation centre and a and a scaling factorscaling factor. .

A A scale factorscale factor is a ratio or number is a ratio or number that represents the amount by which that represents the amount by which a figure is enlarged or reduced: a figure is enlarged or reduced:

(image measurement) ÷ (pre-image (image measurement) ÷ (pre-image measurement)measurement)

Transformations on a Transformations on a Cartesian GridCartesian Grid A A mapmap associates each point of a associates each point of a

geometric shape with a geometric shape with a corresponding point in another corresponding point in another geometric shape on a Cartesian geometric shape on a Cartesian Grid.Grid.

A map shows how a transformation A map shows how a transformation changes a pre-image to an image. changes a pre-image to an image.

An example of a map An example of a map

(2,3) (2,3) → (4,7) means that the point → (4,7) means that the point (2,3) (2,3) maps ontomaps onto point (4,7). point (4,7).

This implies that there is a This implies that there is a relationship between the 2 points.relationship between the 2 points.

(2,3) and (4,7) are called (2,3) and (4,7) are called corresponding pointscorresponding points..

Mapping RuleMapping Rule

The relationship between 2 The relationship between 2 corresponding points, expressed corresponding points, expressed as algebraic expressions, is as algebraic expressions, is called a called a mapping rulemapping rule..

For example: (2,3) → (4,7) has a For example: (2,3) → (4,7) has a mapping rule (x,y) → (x+2, y+4)mapping rule (x,y) → (x+2, y+4)

Properties of Properties of TransformationsTransformations The properties of translations, The properties of translations,

reflections and 180reflections and 180° rotations ° rotations were discussed in Grade 8. were discussed in Grade 8.

These properties are summarized These properties are summarized on the worksheet (GS BLM 6.2 on the worksheet (GS BLM 6.2 Properties of Transformations Properties of Transformations Table)Table)

Minimum Sufficient Minimum Sufficient Conditions for Conditions for TransformationsTransformations To be certain that 2 shapes have To be certain that 2 shapes have

undergone a specific undergone a specific transformation, one must provide transformation, one must provide a minimum sufficient condition a minimum sufficient condition (information). (information).

The Minimum Sufficient The Minimum Sufficient Condition for a Condition for a TranslationTranslation The line segments joining The line segments joining

corresponding points are corresponding points are congruent, parallel and in the congruent, parallel and in the same direction. same direction.

Minimum Sufficient Minimum Sufficient Condition for a ReflectionCondition for a Reflection

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

A perpendicular A perpendicular bisectorbisector A A perpendicular bisectorperpendicular bisector is a line is a line

drawn perpendicular (at a 90drawn perpendicular (at a 90° ° angle) to a line segment dividing angle) to a line segment dividing it into 2 equal parts.it into 2 equal parts.

The perpendicular bisector always The perpendicular bisector always intersects with the intersects with the midpointmidpoint of of the original line segment. the original line segment.

Minimum Sufficient Minimum Sufficient Condition for a 180Condition for a 180° ° RotationRotation The line segments joining The line segments joining

corresponding points intersect at corresponding points intersect at their midpoints. their midpoints.

Regular polyhedron Regular polyhedron (Grade 7)(Grade 7) A A regular polyhedronregular polyhedron is a 3-D is a 3-D

figure with faces that are figure with faces that are polygons.polygons.

Polyhedron’s plural is polyhedraPolyhedron’s plural is polyhedra..

Platonic solids Platonic solids

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

The 5 Platonic solids The 5 Platonic solids

1.1. The cubeThe cube

2.2. The regular tetrahedronThe regular tetrahedron

3.3. The regular octahedronThe regular octahedron

4.4. The regular dodecahedronThe regular dodecahedron

5.5. The regular icosahedronThe regular icosahedron See Page 39 of Math 9 bookletSee Page 39 of Math 9 booklet

The 3 characteristics of The 3 characteristics of regular polyhedra (Platonic regular polyhedra (Platonic solids)solids)

1.1. All faces are 1 type of regular All faces are 1 type of regular polygon.polygon.

2.2. All faces are congruent.All faces are congruent.

3.3. All vertices are the same (i.e. All vertices are the same (i.e. they have they have vertex regularityvertex regularity))

What is vertex What is vertex regularity?regularity? When all vertices in a polyhedron are When all vertices in a polyhedron are

the same, you have the same, you have vertex vertex regularityregularity, which can be described , which can be described using notation. using notation.

For example, the notation {5,5,5} For example, the notation {5,5,5} represents the vertex regularity of a represents the vertex regularity of a regular dodecahedron because there regular dodecahedron because there are 3 regular 5-sided polygons at are 3 regular 5-sided polygons at every vertex.every vertex.

Circle GeometryCircle Geometry

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

Central anglesCentral angles Inscribed anglesInscribed angles Tangent of a Tangent of a

circlecircle Circumscribed Circumscribed

angleangle

Central angleCentral angle

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

Inscribed angleInscribed angle

An An inscribed angleinscribed angle is an angle is an angle that has its vertex on a circle and that has its vertex on a circle and is is subtendedsubtended by an arc of the by an arc of the circle. (page 42)circle. (page 42)

What does “subtended” mean What does “subtended” mean geometrically?geometrically?

Tangent of a circleTangent of a circle

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

Circumscribed angleCircumscribed angle

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

A polygonA polygon

A A polygonpolygon has all sides congruent has all sides congruent and all angles congruent. and all angles congruent.

Polygons can be both Polygons can be both regularregular and and irregularirregular..

Regular polygons have both Regular polygons have both reflective and rotational reflective and rotational symmetry. (Major difference symmetry. (Major difference between regular and irregular between regular and irregular polygons)polygons)

Regular polyhedronRegular polyhedron

A A regular polyhedronregular polyhedron is a 3-D is a 3-D figure with faces that are figure with faces that are polygons.polygons.

Polyhedron’s plural is polyhedraPolyhedron’s plural is polyhedra..

Polyhedra with regular Polyhedra with regular polygonal facespolygonal faces In grade 9 In grade 9

Geometry, there Geometry, there are several types are several types of polyhedra: of polyhedra:

The 5 Platonic The 5 Platonic solidssolids

A uniform prismA uniform prism An antiprismAn antiprism A deltahedronA deltahedron A dipyramidA dipyramid The Archimedean The Archimedean

solidssolids

The 5 Platonic solidsThe 5 Platonic solids

1.1. The cubeThe cube

2.2. The regular tetrahedronThe regular tetrahedron

3.3. The regular octahedronThe regular octahedron

4.4. The regular dodecahedronThe regular dodecahedron

5.5. The regular icosahedronThe regular icosahedron See Page 39 of Math 9 bookletSee Page 39 of Math 9 booklet

Uniform prismUniform prism

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

AntiprismAntiprism

An An antiprismantiprism is a polyhedron is a polyhedron formed by 2 parallel, congruent formed by 2 parallel, congruent bases and triangles. bases and triangles.

Each triangular face is adjacent Each triangular face is adjacent (next to) 1 of the congruent (next to) 1 of the congruent bases. bases.

Page 51Page 51

DeltahedronDeltahedron

A A deltahedrondeltahedron is a polyhedron is a polyhedron that has only equilateral triangle that has only equilateral triangle faces. faces.

The deltahedron is named after The deltahedron is named after the Greek symbol delta (the Greek symbol delta (ΔΔ))

The plural is deltahedra.The plural is deltahedra. Page 51Page 51

DipyramidDipyramid

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

Page 52Page 52

Archimedean solidsArchimedean solids

The Archimedean solids are the The Archimedean solids are the 13 different semi-regular 13 different semi-regular polyhedra. polyhedra.

The Archimedean solids have The Archimedean solids have vertex regularity and symmetry vertex regularity and symmetry (reflective and rotational)(reflective and rotational)

13 Archimedean solids 13 Archimedean solids (page 53)(page 53) CuboctahedronCuboctahedron Great rhombicosidodecahedronGreat rhombicosidodecahedron Great rhombicuboctahedronGreat rhombicuboctahedron IcosidodecahedronIcosidodecahedron Small rhombicosidodecahedronSmall rhombicosidodecahedron Small rhombicuboctahedronSmall rhombicuboctahedron Snub cubeSnub cube

13 Archimedean solids 13 Archimedean solids (page 53) continued(page 53) continued Snub dodecahedronSnub dodecahedron Truncated dodecahedronTruncated dodecahedron Truncated icosahedronTruncated icosahedron Truncated octahedronTruncated octahedron Truncated tetrahedronTruncated tetrahedron Truncated cubeTruncated cube

What is a vertex?What is a vertex?

A A vertexvertex is a point at which 2 or is a point at which 2 or more edges of a figure meet. more edges of a figure meet.

The plural is vertices. The plural is vertices.

Vertex configurationVertex configuration

Vertex configurationVertex configuration is the arrangement is the arrangement of regular polygons at the vertices of a of regular polygons at the vertices of a polyhedron. (page 50)polyhedron. (page 50)

Vertex configuration notation refers to the Vertex configuration notation refers to the types of regular polygons around a types of regular polygons around a vertex.vertex.

For example, the notation {3,4,5,4} For example, the notation {3,4,5,4} means that a vertex has an equilateral means that a vertex has an equilateral triangle, a square, a regular pentagon triangle, a square, a regular pentagon and a square around it in that order. and a square around it in that order.

Plane of symmetryPlane of symmetry

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

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Axis of symmetryAxis of symmetry

An An axis of symmetryaxis of symmetry is a line is a line about which a polyhedron about which a polyhedron coincides with itself as it rotates.coincides with itself as it rotates.

The number of times a The number of times a polyhedron coincides with itself in polyhedron coincides with itself in 1 complete rotation is its 1 complete rotation is its order of order of rotational symmetryrotational symmetry. .

The properties of The properties of regular polyhedraregular polyhedra1.1. All faces are regular polygons.All faces are regular polygons.2.2. All faces are the same type of All faces are the same type of

congruent polygon.congruent polygon.3.3. The same number of faces meet at The same number of faces meet at

each vertex. each vertex. 4.4. Regular polyhedra have several axis Regular polyhedra have several axis

of symmetry (rotational symmetry)of symmetry (rotational symmetry)5.5. Regular polyhedra have several Regular polyhedra have several

planes of symmetry (reflective planes of symmetry (reflective symmetry)symmetry)

The difference between The difference between semi-regular and regular semi-regular and regular polyhedrapolyhedra Regular polyhedra = Platonic Regular polyhedra = Platonic

solids, etc.solids, etc. Semi-regular polyhedra = Semi-regular polyhedra =

Archimedean solidsArchimedean solids All faces of a semi-regular All faces of a semi-regular

polyhedron are not the same type polyhedron are not the same type of regular polygon. of regular polygon.