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. 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 Common prefixes

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. 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 The prefixes of the metric system

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 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.

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)*(Π)*(r) or C=(Π)*(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 = (Π)*(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. There are 3 common examples: a rectangular prism a cylinder a triangular prism Examples of prisms and cylinders

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 prism’s base x prism’s height • Vprism = Abase x h

35. How to calculate the volume of a cylinder: • To calculate the volume of a cylinder: • Vcylindre = Πr2 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. We are going to calculate the volume of three 3-D figures: A cone A pyramid A sphere 3-D figures

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 Πr2 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 Πr3

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= 2Πr2 + 2Πrh

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 = Πr2 + Πro

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 = 4Πr2

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