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

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.

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.

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)

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)

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)

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

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)

Examples of common regular polygons Regular trigon (equilateral triangle) Regular tetrahedron (square) Regular pentagon Regular hexagon Regular octagon

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

Converting measurements into different metric units To convert metric units, you must use the metric staircase.

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

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

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

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

The perimeter The perimeter is the total distance around a figure’s 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.

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.

Area of a rectangle To calculate the area of a rectangle: Arectangle = length x width

Area of a triangle To calculate the area of a triangle: Atriangle = ½ x base x height

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.

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.

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.

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.

How to calculate the circumference To calculate the circumference of a circle: C = (2)*(Π)*(r) or C=(Π)*(d) r is the radius of the circle d is the diameter of the circle.

How to calculate the area of a circle

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.

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

Examples of prisms and cylinders There are 3 common examples: a rectangular prism a cylinder a triangular prism

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.

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

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)

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.

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.

How to calculate the volume of a prism: Vprism = area of the prism’s base x prism’s height Vprism = Abase x h

How to calculate the volume of a cylinder: Vcylindre = Πr2 x h

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.

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.

3-D figures We are going to calculate the volume of three 3-D figures: A cone A pyramid A sphere

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

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 Πr2 x h

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.

How to calculate the volume of a pyramid Vpyramide= 1/3 x (the volume of prism) Vpyramide = 1/3 x Abase x h

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.

How to calculate the volume of a sphere Volume of a sphere = 4/3 x Πr3

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

How to calculate the surface area of a cylinder At= 2Πr2 + 2Πrh

How to calculate the surface area of a cone It is the sum of the base area and the lateral area. At = Πr2 + Πro

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

How to calculate the surface area of a sphere At = 4Πr2

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

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

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.

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

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.

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.

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.

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

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.

The types of transformations There are 4 types of transformations: Translations Reflections Rotations Dilatations.

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

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

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

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)

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.

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.

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) → (x+2, y+4)

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)

Minimum Sufficient Conditions for Transformations To be certain that 2 shapes have undergone a specific transformation, one must provide a minimum sufficient condition (information).

The Minimum Sufficient Condition for a Translation The line segments joining corresponding points are congruent, parallel and in the same direction.

Minimum Sufficient Condition for a Reflection The line segments joining corresponding points have a common perpendicular bisector.

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.

Minimum Sufficient Condition for a 180° Rotation The line segments joining corresponding points intersect at their midpoints.

Regular polyhedron (Grade 7) A regular polyhedron is a 3-D figure with faces that are polygons. Polyhedron’s plural is polyhedra.

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

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

The 3 characteristics of regular polyhedra (Platonic solids) All faces are 1 type of regular polygon. All faces are congruent. All vertices are the same (i.e. they have vertex regularity)

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.

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

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

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?

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)

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

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)

Regular polyhedron A regular polyhedron is a 3-D figure with faces that are polygons. Polyhedron’s plural is polyhedra.

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

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

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

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

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

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

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

13 Archimedean solids (page 53) Cuboctahedron Great rhombicosidodecahedron Great rhombicuboctahedron Icosidodecahedron Small rhombicosidodecahedron Small rhombicuboctahedron Snub cube

13 Archimedean solids (page 53) continued Snub dodecahedron Truncated dodecahedron Truncated icosahedron Truncated octahedron Truncated tetrahedron Truncated cube

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

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.

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

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.

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

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.