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M2 Geometry Journal 1: Chapter 1 Geometry

M2 Geometry Journal 1: Chapter 1 Geometry. José Antonio Weymann. Point: A single dot in space, used to describe location. It is described with a dot and a capital letter. P Line: A straight connection of points that goes on forever in both directions.

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M2 Geometry Journal 1: Chapter 1 Geometry

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  1. M2 Geometry Journal 1: Chapter 1 Geometry José Antonio Weymann

  2. Point: A single dot in space, used to describe location. It is described with a dot and a capital letter. P Line: A straight connection of points that goes on forever in both directions. XY YX Plane: A flat surface that goes on forever, it has no thickness and it contains 3 points. POINT,LINE and PLANE T

  3. The difference between collinear and coplanar points is collinear points coexist in in the same line and coplanar points coexist in the same plane. Collinear Coplanar: P N A B M Collinear and Coplanar points

  4. Collinear and Coplanar points Collinear Counterexapmple Coplanar Counterexample

  5. Line: A straight connection of points that goes on forever in both directions. XY YX Segment: Any straight collection of dots that has a beginning and an end (endpoints) XY Ray: A straight collection of points that has one end point and goes on forever. XY *This are related because they all are ways to use points and lines, also we will be using them for the rest of the school year. Besides their everyday location and distance applications. line, segment, and ray

  6. It is when two lines intersect through the same line in any situation. Like in Postulates 1-1-4 (If two lines intersect, then they intersect in exactly one point) and 1-1-5 (If two planes intersect, then they intersect in exactly one line) Intersection of lines and planes

  7. The Difference between Postulate, Axiom and Theorem is between Postulate and Axiom is nothing they are interchangeable terms (this are accepted truths as fact with out proof), but between this and theorem is that a theorem is a theoretical proposition, later to be proved by other propositions and formulas. postulate, axiom and theorem

  8. To measure segments use a ruler and just subtract the values at the end points. A Field is 120 ft long and a player starts running from 60 ft , what is the distance when he runs to point 120 ft? Answer: 60 ft, The road is 130km long and a car starts its trip from south park which is 90km up the road, when he reaches the 130 km line how many km does the car have traveled? Answer: 40km The sidewalk is 850mt long and an old lady starts walking from 500mt, when she reaches the end of the sidewalk, what’s her current distance Answer: 350mt. Ruler Postulate (postulate 1-2-1)

  9. If A,B,C (our three collinear points) and B is between A & C, then AB+BC=AC 24 46 C A B 24+46= 70, so CB = 70 so CA+AB= CB 72 20 A B C 72+20= 92, so AC = 92 so AB+BC=AC 75 55 E F G 75+55= 130, so EG = 130 so EF+FG=EG Segment Addition postulate (1-2-2)

  10. To find the distance between two points on a coordinate plane you have to take the X1 and X2 coordinates and square them then the Y1 & Y2 coordinates and do the same, add and then square the answer. d = √ (X₁-X₂)₂+(Y₁ -Y₂)₂ AB= √(5-0)₂+(1-3)₂ * CD= √(-3 –(-1)₂+(-4 - 1)₂ √5₂ + (-2)₂ √(-2)₂+(-5)₂ √25+ 4 √ 4+25 This are congruent √29 √29 d= √(4-1)₂+(2-6)₂ √3₂+(-4)₂ √9+16 √25 = 5 this is a random problem distance between two points on a coordinate plane

  11. Congruence means and equal measure, not necessarily a value, we are comparing names; this contrasts with equality because that means 2 things with same value, therefore it is comparing values. They are similar because both are comparing two things. They are different because congruent is two objects exactly the same regardless of orientation, and equality is shape size and angles, like two squares they are congruent but don’t have measurements meaning we don’t know if they are equal. ≈ = Congruence and Equality

  12. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a₂+b₂=c₂ a=4 and b=5 a=3 and b=4 find the value of c c₂= a₂+b₂ c₂= a₂+b₂ a₂+b₂=c₂ a=4     b=5 =4₂+5₂ = 3₂+4₂ 4₂+5₂= c₂ =16+25 =9+16 16+25= c₂ =41 =25 41= c₂ c=√41 c=√25 c≈ 6.4 c≈ 5 Pythagorean theorem (theorem 1-6-1)

  13. Angles are 2 rays that share a common end point. There are acute (smaller than 90°), right (90°), obtuse (bigger than 90°) and finally straight angles (180°) Right Acute Obtuse Straight Exterior Interior angles

  14. 2 small angles add up to the big angle. 1.angle ABD=150 2.angle EFI=180 3.angle VXY=110 angle ABC=45 angle FGI=90 angle YXZ=60 150-45= 180-90= 110+60= CBD=105 EFG=90 VXY=170 Angle Addition postulate (1-3-2)

  15. Midpoint is what we call the middle of the segment, equidistant from the endpoints; cuts into two equal smaller segments. To construct a midpoint first draw a segment, second draw a line a little bit passed the approximate middle, third draw two arcs from both endpoints in the two sides of the segment, fourth draw a straight line through the crossings and you have your constructed midpoint. MIDPOINT FORMULA: (X₁+X₂/2,Y₁+Y₂/2) 1.(6+4)(4+1)/2 m=1.5 2.(-3+0)(-1,1)/2 m=-1.5 3. (-3+-1.5)(4+1)/2 m= -.75,2.5 Midpoint (construction and midpoint formula)

  16. An angle bisector is a line which cuts and angle into two equal parts, therefore the word bisect means to cut in have. First lock your compass, make an arc in each side put the point in each arc and make an arc in the interior; then connect the vertex to the intersection point. Angle bisector (construction)

  17. Adjacent: are 2 angles that have the same vertex and a same side. Linear Pairs of Angles: 2 adjacent straight angles that will form a straight line. Meaning all Linear Pairs of Angles are supplementary (Linear Pair Postulate, L.P.P) Vertical: Non adjacent angles formed when 2 lines intersect; vertical angles are always congruent. Adjacent: Linear Pairs: Vertical: Adjacent, vertical and linear pairs

  18. Complementary angles are 2 angles that add up to 90º and supplementary add up to 180º; supplementary are always linear pairs but complementary aren`t, they are always adjacent. Complementary: 90º Supplementary: 180º Complementary and supplementary

  19. Square: to take the perimeter of a square add all 4 sides (P=4s) and area just square one side (A=s₂) Rectangle: to take the perimeter of a rectangle add the 2 lengths and 2 widths (P=2l+2w) and area multiply length time width (A=lw) Triangle: to take the perimeter of a triangle (a+b+c) and to take the area (bh/2) Square: P=4(4), P=16ft. Rectangle: 5ft. 4ft. A=4₂, A= 16ft₂ 4ft. P=5+5+4+4, P=18ft. A=5*4, A=20ft.₂ Perimeter And Area

  20. Triangle: P= 2x+3x+5+10 A=1/2(3x+5)(2x) = 5x+15 =3x₂+5x 2nd round of examples: Square: Rectangle: 3cm. P=6(4) P=24 5cm. A= 6₂ A= 36 P=3+3+5+5 P= 16cm. A=3*5 A=15cm. ₂ Triangle: P=4x+5x+3+9 A=1/2(5x+9)(4x) = 9x+12 =5x₂+5x Perimeter and Area

  21. The circumference of a circle is the distance around the circle. Circumference (C) is given by C=(Pi)d or C=2(Pi)r. C=2(Pi)r C=2(Pi)r =2(Pi)(3)=6(Pi) =2(Pi)(11) ≈ 18.8cm =22(Pi) ≈69.1cm. 3cm. 11cm. Circumference of a circle

  22. A = (Pi)r₂ A= (Pi)r₂ = (Pi)(3)₂ = 9(Pi) 3cm. ≈ 28.3cm₂ A= (Pi)r₂ = (Pi)(11)₂ = 121(Pi) ≈380.1cm₂ 11cm. Area of a circle

  23. 1.READ IT CAREFULLY2.WRITE DOWN ALL IMPORTANT INFORMATION3.DRAW A PICTURE4.WRITE AND SOLVE THE EQUATION5.ANSWER THE QUESTION Five step problem solving Process

  24. 1. Read it Carefully: The quilt pattern includes 32 small triangles. Each has a base of 3 inches and a height of 1.5 in. Find the amount of fabric used to make the 32 triangles. 2. Write down all important information: 3 inches, 32 small triangles, 1.5 inches 3. Draw a Picture h:1.5in. b: Five step problem solving Process

  25. 4. Write and Solve the equation b: 3in. h:1.5in. bh/2 (3)(1.5)/2 *32 = 72 5. Answer the Question Answer: The amount of fabric used to make the 32 triangles in the quilt is 72in₂ Five step problem solving Process

  26. Things to know before entering transformations Pre-image: Image: A ABC A’B’C’ Introduction to Transformations A

  27. A Transformation is a change in the position of the original object. Translation : The slide of an object in any direction (usually is the most common) (X,Y) (X+a,Y+b) transformations

  28. Rotation: to rotate a figure around a point. Transformations

  29. Reflection: When you mirror your Pre-image across any line. If across Y-axis (X,Y)(-X,Y). If across X-axis (X,Y)(X,-Y) Transformations

  30. This geometry chapter 1 journal has ended, good luck on the quiz and I hope to have covered the topics In a satisfactory matter. Journal Has been concluded

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