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GEOMETRY Presented by Umaira Bukhari The law of nature are about the mathematical thoughts of God ”---Euclid :
The law of nature are about the mathematical thoughts of God ”---EuclidWhat is the Geometry?:
What is the Geometry? Geometry ( Greek γεωμετρί α; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships . Geometry was one of the two fields of pre-modern mathematics , the other being the study of numbers ( arithmetic ). Euclid as a Father of Geometry:
Euclid as a Father of Geometry Classic geometry was focused in compass and straightedge constructions . Geometry was revolutionized by Euclid , who introduced mathematical rigor and the axiomatic method still in use today His book, The Elements is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th centuryPowerPoint Presentation:
In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry.Did humans invent geometry??:
Did humans invent geometry?? Absolutely not!!!! Human beings didn’t invent geometry, we discovered them. What is geometry Really about? :
What is geometry Really about? Pictures or visual patterns. Construction TheoremsActivity 1:
Activity 1 Draw each figure without lifting pen from paper or retracting line. Geometry in Nature:
Geometry in NatureSymmetry:
Symmetry Definition -an attribute of a shape or relation; exact reflection of form on opposite sides of a dividing line or plane. Types of SymmetrySymmetry in Nature:
Symmetry in Nature Symmetry can be Found All Around Us.Reflective Symmetry:
Reflective Symmetry Also known as line symmetry, means that one half of an image is the mirror image of the other half.Point Symmetry:
Point Symmetry Point symmetry - any straight cut through the center point divides the organism into mirroring halves.Reflective Symmetry:
Reflective Symmetry Another example of this particular symmetry in nature, is a reflection on the water .Rotational Symmetry:
Rotational Symmetry The planets, with slight variation due to chance, exhibit radial symmetry.Polygons:
Polygons Polygons are closed plane figures made up by 3 or more connecting line segments.Polygons in Nature:
Polygons in Nature There are polygons found in nature everywhere , you just have to take a closer look! Have you ever stopped to consider how many inanimate things that we see in nature that are geometrically arranged?Polygons found in Fruit:
Polygons found in Fruit If you slice a kiwi in half, you will see that the core forms a six-sided shape, also known as a hexagon. Look closely at a pineapple and you will see that all pineapples have the same skin, they are tessellations of trapezoids. This is also true for an apple, except it is a slightly different version of a pentagon, it becomes a star.Animals and Polygons:
Animals and Polygons Most polygons found on animals are repetitive, but slightly altered. On these two cheetahs, almost all of the polygons are present, except a solid triangle . Have you ever looked closely at a honeycomb? Each cell wall stands at a correct 120 angle with respect to one another to form a tessellation of regular hexagons. A giraffe’s body is completely covered in regular polygons .On your own time:
On your own time N ext time you step outside, take a look around you and see what polygons you can find within nature itself. You might be surprised as to what you actually do find!Tessellations:
Tessellations Definition- repeating pattern of distinct shapes Regular Tessellations- tessellations of only one type of polygon Semi-regular Tessellation- tessellation of more than one polygon.Examples of Tessellations:
Examples of Tessellations Semi-regular Tessellation Regular TessellationExamples of Tessellations in Nature:
Examples of Tessellations in Nature Division of CellsMore Examples:
More Examples Honey Comb Fish scales Shell of a turtle Pineapple Ear of cornSpheres in Nature:
Spheres in Nature Geometry Geo-Earth Metry-measurement Angles Great CircleThe Earth:
The Earth Sphere Lines of latitude and longitude EquatorOther Examples of Spheres:
Other Examples of Spheres Sun Moon Planets OrangesGeometry in Daily Life:
Geometry in Daily LifeGeometry in Architecture:
Geometry in Architecture Arches are used to withstand maximum weight. Structural designs to withstand forces of natureGeometry in Photography:
Geometry in Photography The reason that 45 degree lighting is so important is that it’s the perfect angle to create modeling on the human form. The term modeling refers to showing three dimensionality through the use of light. When you have light coming from where the camera is, that three dimensionality is lost because shadows aren’t seen on the face. Put the light off the camera and you get shadows, which gives you 3D, and 45 degrees is the perfect angle to maximize this effect. Camera Lighting SubjectPowerPoint Presentation:
60 o Projectile Range Stairs : inclined at 60 o , with each stair at 90 o . 30 o 30 o 120 o A Simple cloth hanger has 2 x 30 o angles + 1 x 120 o angle = 180 o angle. Semicircles in CyclesGeometry in Nature:
Geometry in Nature Symmetry of a Leaf Geometry of Lunar EclipsePowerPoint Presentation:
90 o 60 o Geometry in Stairs . Inclined Angle = 60 o angle; Each Stairs = 90 o anglePowerPoint Presentation:
Geometry in Cycles . Racing bikes are made using best geometry to give maximum efficiency .PowerPoint Presentation:
The shortest path between two points is a straight line. The two sides of Roads are always parallel to each otherPowerPoint Presentation:
Specific examples Geometry in Cycles . Racing bikes are made using best geometry to give maximum efficiency. 90 o 60 o Geometry in Stairs . Inclined Angle = 60 o angle; Each Stairs = 90 o angle:
A REVIEW OF BASIC CONCEPTSTable of Contents:
Table of Contents Points Line Segments Lines Rays AnglesPOINT:
POINT A point is a location in space. Capital letters are used to name points. A K G MLINE SEGMENT:
LINE SEGMENT A line segment is made up of 2 points and the straight path between them. N D The two points at the ends of a line segment are called endpoints . This line segment is called: ND or DNQuiz Me!:
Quiz Me! 1. What is the name of this geometric figure? B. Line Segment A. Line C. Ray M E Table of Contents BACKFabulous!:
Fabulous! What is the name of this geometric figure? B. Line Segment M E You remembered a line segment is made up of 2 points and the straight path between them! Table of ContentsLINE:
LINE A line is a straight path that goes on forever in both directions. J R The arrowheads symbolize the line continuing forever, even though it looks like it stops at the arrowheads. This line is called: RJ or JR R and J are points , not end points! Table of ContentsQuiz Me!:
Quiz Me! What is the name of this geometric figure? A. Line B. Line Segment C. Ray Table of Contents H Z BACKTry Again:
Try Again What is the name of this geometric figure? A. Line B. Line Segment C. Ray H Z Need a little help? BACKSuperb!:
Superb! What is the name of this geometric figure? A. Line Table of Contents H Z You remembered a line is a straight path that goes on forever in both directions!RAY:
RAY A ray is a straight path that has a starting point and goes on forever in one direction . B W This is point B This is endpoint W This ray is called: WB Note the difference between point and endpoint! The endpoint is always the first letter in the name of a ray!Quiz Me!:
Quiz Me! 3. What is the name of this ray? D O A. OD C. DO B. OD Table of Contents BACKFantastic!:
Fantastic! 3. What is the name of this ray? D O A. OD You remembered that the endpoint is always the first letter in the name of a ray! Table of ContentsANGLE:
ANGLE NEXT An angle is formed by 2 rays or 2 line segments that share the same endpoint. The endpoint where the rays or segments meet is called the vertex of the angle. The rays or segments are called the sides of the angle. Table of ContentsNaming Angles:
Naming Angles The vertex of the angle must always be in the middle of the, between the points on the sides . A B C This angle is called: ABC The vertex point is B. That is why B is in the middle of the sides A and C. CBA or Table of ContentsClassifying Angles:
Classifying Angles Straight Angle Measures exactly 180 Acute Angle Measures between 0 and 90 Reflex Angle Measures between 180 and 360 Obtuse Angle Measures between 90 and 180 Right Angle Measures exactly 90 BACKQuiz Me!:
Quiz Me! Table of Contents 4. What is the classification of this angle? A. Acute Angle B. Reflex Angle C. Obtuse Angle BACKAwesome!:
Awesome! 4. What is the classification of this angle? C. Obtuse Angle You remembered that an obtuse angle measures between 90 and 180 !Geometry Song:
Geometry Song