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Section One

Section One

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Section One

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  1. Section One Patterns and Inductive Reasoning

  2. Inductive Reasoning • Reasoning that is based on patterns you observe. Find a pattern: 3,6,12,24,… Find a pattern: 384,192,96,48,…

  3. Conjecture • A conclusion you reach using inductive reasoning • Make a conjecture about the sum of the first 30 odd numbers: • = 1 • 1 + 3 = 4 • 1 + 3 + 5 = 9 12 22 32 You can conclude that the sum of the first 30 odd numbers is 30² = 900

  4. Conjecture • A conclusion you reach using inductive reasoning Make a conjecture about the sum of the first 8 even numbers: 2 = 1 * 2 2 +4 = 2 * 3 2 + 4 + 6 = 3 * 4 Therefore, the first 8 even numbers = 8 * 9 = 72 You can conclude that the sum of the first 50 even numbers is 50 * 51 = 2550

  5. Find one counterexample: The sum of two numbers is greater than either number 8 + (-5) = 3 3 > 8 Counterexample • Not all conjectures turn out to be true. • If you can find one example that proves the conjecture false, then you have found a counterexample

  6. Closure • How can you use a conjecture to help solve a problem? A conjecture can be tested to see whether it is a solution

  7. Section Two Points, Lines, and Planes

  8. How many lines can you draw? Mark 3 points. How many lines can you draw? 3 Lines Mark 4 points. How many lines can you draw? 6 Lines Mark 5 points. How many lines can you draw? 10 Lines Mark 6 points. How many lines can you draw? 15 Lines

  9. Definitions Point – has no size; it is a location represented by a small dot and is named by a capital letter ( a geometric figure is a set of points) Line – a series of points that extends in two opposite directions without end Collinear – points that lie on the same line

  10. D C • A • B • Plane – is a flat surface that has no thickness. A plane contains many lines and extends without end in the directions of its lines Plane D Coplanar – points and lines in the same plane Since it takes 3 points to define a plane, 3 points are always coplanar. Unlike 4 or more points are sometimes coplanar. Plane ABC

  11. Postulate (or axiom)- an accepted statement of fact Postulate 1-1: Through any two points is exactly one line Line t is the only line that passes through points A and B t • B • A Postulate 1-2 : If two lines intersect, then they intersect in exactly one point • • B E AE and BD intersect at C C • • A D

  12. Postulate 1-3: If two planes intersect,then they intersect in exactly one line R • S • T • W Plane RST and Plane STW intersect in line ST Postulate 1-4: Through any three noncollinear points there is exactly one plane

  13. Closure How are a postulate and conjecture alike? You accept a postulate as true without proof How are a postulate and conjecture different? You try to determine whether a conjecture is true or false

  14. Section Three Segments, Rays, Parallel Lines and Planes

  15. Segment   Segment AB Segment – the part of a line consisting of two endpoints and all points between them A B Ray – part of a line consisting of one endpoint and all the points of the line on one side of the endpoint   Ray XY X Y Opposite Rays – two collinear rays with the same endpoint. Opposite rays always form a line.    ML and MN are opposite rays L M N

  16. Skew lines –are not parallel and do not intersect, therefore, they are noncoplanar Parallel lines – are coplanar lines that do not intersect E AB and CD are parallel lines AB and EF are skew lines A B F C D

  17. Parallel Planes – are planes that do not intersect L A P Y C I N E Plane PLAY is parallel with plane NICE

  18. Closure How are parallel lines and skew lines alike? How are parallel lines and skew lines different? Both parallel lines and skew lines never intersect Parallel lines are coplanar and skew lines are noncoplanar

  19. Section 1.4 Measuring Segments and Angles

  20. Postulate 1-5 The points of a line can be put into one-to-one Correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers a b A B The length of AB = a - b

  21. Two segments with the same length are CONGRUENT SEGMENTS A B C D AB = CD therefore AB  CD

  22. Postulate 1-6 If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC A B C

  23. Midpoint – of a segment is a point that divides a segment into two congruent segments; it bisects the segment

  24. Angle – an angle is formed by two rays with the same endpoint. The endpoint is the vertex of the angle. A B C Ray BA and Ray BC for angle  ABC B is the endpoint. B is the vertex of  ABC

  25. Postulate 1-7: Protractor Postulate Let OA and OB be opposite rays in a plane. OA, OB, and all the rays with endpoint O that can be drawn on one side of AB can be paired with the real numbers from 0 to 180 so that a. OA is paired with 0 and OB is paired with 180 b. If OC is paired with x and OD is paired with y, then m COD = |x - y|

  26. Classify angles to their measures Acute angle 0 < X < 90 Right Angle X = 90 Obtuse Angle 90 < X < 180 Straight Angle X = 180

  27. Postulate 1-8: Angle Addition If point B is in the interior of AOC, then m AOB + m BOC = m AOC If AOC is a straight angle, then m AOB + m BOC = 180 A B O C B O A C

  28. Closure • How can two of the postulates in this lesson help you measure segments and angles? The segment add. Postulate states that the sum of all the parts of a segment equals the whole segment. Angle add. Postulate states that an angle can be divided into two angles, the sum of whose measures equals the measure of the whole angle

  29. Definitions • Perpendicular lines – two lines that intersect to form right angles. • Perpendicular bisector of a segment – is a line, segment, or ray that is perpendicular to the segment at its midpoint, bisecting the segment into two congruent segments • Angle bisector – a ray that divides an angle into two congruent coplanar angles. Its endpoint is at the angle vertex.

  30. Section 1.6 The Coordinate Plane

  31. A B Distance Formula The distance dbetween two points A(x₁, y₁) and B(x₂, y₂) is calculated using the formula below: d =  (x₂ – x₁)² + (y₂ – y₁) ² (x1 , y1) (x2 , y2)

  32. d =  ((x₂ – x₁)² + (y₂ – y₁)²) d =  ((-8 – 10)² + (14 – 14 )²) d =  ((-18)² + (0)²) d =  324 + 0 Examples L (10,14) M (-8, 14) d = 18

  33. d =  ((x₂ – x₁)² + (y₂ – y₁)²) d =  ((-8 – -3)² + (6 - -2)²) d =  ((-5)² + (8)²) d =  25 + 64 Examples L (-3, -2) M (-8, 6) d = 9.43

  34. The Midpoint Formula (mx, my) The (x, y) coordinates of the midpoint M, of line segment AB whose endpoints are A(x₁, y₁) and B(x₂, y₂), are found using this formula Coordinates of Mx and My are: x₁+ x₂ , y₁ + y₂ 2 2

  35. Coordinates of Mx and My are: Mx = x₁+ x₂ and My = y₁ + y₂ 2 2 If given one endpoint and the midpoint, we can find the (x,y) coordinates of the other endpoint by simplifing this formula into: Mx = x2 + x1 My = y2 + y1 2 2 2Mx = x2 + x1 2My = y2 + y1 2Mx – x1 = x2 2My – y1 = y2

  36. Closure If you know the coordinates of the midpoint and one endpoint, can you find the coordinates of the other endpoint? 2Mx – x1 = x2 2My– y1 = y2 Midpoint: (2,5) Endpoint A (-1,5) Endpoint B (x, y)

  37. d =  ((5 - -6)² + (-1 - -6)²) d =  ((11)² + (5)²) d =  (121 + 25) d =  146 Finding the distance and midpoint of a geometric figure (-2 , 8) d2 d3 d1 = 12.08 (5, -1) d1 d2 = 11.40 (-6, -6) d3 = 14.56

  38. Section 1.7 Perimeter, Circumference, And Area

  39. Perimeter • The perimeter of a polygon is the sum of the lengths of its sides Perimeter of a square = side + side + side + side = 4s S If the side of a square is 8cm, what is the perimeter? S S S

  40. Perimeter • Perimeter of a rectangle = base + base + height + height = 2b + 2h B If the base is 8 inches and the height is 4 inches, what is the perimeter? H H B

  41. Area • The area of a polygon is the number of square units the figure encloses Area of a square = (side)(side) = s What is the area of the square??? 4 ft 4 ft

  42. Area Area of a rectangle = (base)(height) = bh 12 in 3 ft What is the area of the rectangle???

  43. 6 in Circumference • This is referred to when measuring the perimeter of a circle Perimeter of a circle = circumference of a circle = πd (or 2πr ) What is the circumference of the circle???

  44. Area of a circle • Area = r2 What if we were given the diameter. How do we use this formula??? 2 radii = 1 diameter Area = r2= (d/2)2 =  d2 4

  45. Closure • Can you find the area and perimeter of the square? Circle? The sides of the square measure 8 cm Square: Area = 64 cm² ; perimeter = 32 cm Circle: Area = 32π cm² ; circumference = 82π cm