Upload
sheilaabad8766
View
235
Download
0
Embed Size (px)
DESCRIPTION
geometry
Citation preview
GEOMETRYGEOMETRYProperties and relationships of points, lines, planes,
solids. Undefined elements: point, line, planeAxiom: self-evident truthPostulate: statement accepted as true without proofTheorem: statement which is proven2 lines are parallel iff they line on same plane but do not
intersect2 planes or a line & a plane are parallel if they do not
intersect2 lines are perpendicular/orthogonal if the adjacent
angles of their point of intersection are right anglesSkew lines: 2 lines not coplanar/non-parallel/non-
intersection
EUCLIDEAN GEOMETRYEUCLIDEAN GEOMETRYElementsDeductive ReasoningPostulates:1. Two points determine a line2. A line can be extended indefinitely in both directions3. A circle may be drawn with any given center and
radius4. All right angles are equal5. Parallel Postulate Given a point P not on a line L, then there is one and
only one line (L2) passing thru P and parallel to L
EUCLIDEAN GEOMETRYEUCLIDEAN GEOMETRYElementsDeductive ReasoningPostulates:1. Two points determine a line2. A line can be extended indefinitely in both directions3. A circle may be drawn with any given center and
radius4. All right angles are equal5. Parallel Postulate Given a point P not on a line L, then there is one and
only one line (L2) passing thru P and parallel to L
HYPERBOLIC GEOMETRYHYPERBOLIC GEOMETRY
Karl Friedrich Gauss Nicholas Lobachevski Johann BolyaiDisc Model : Hyperbolic planeFundamental Circle: Fixed circle C, center at OHyperbolic points: interior to CHyperbolic lines: circular arc perpendicular to C
HYPERBOLIC GEOMETRYHYPERBOLIC GEOMETRY
Disc Model : Hyperbolic planeFundamental Circle: Fixed circle C, center at O
(Poincare Disc)Hyperbolic points: interior to circle CHyperbolic lines: circular arc perpendicular to circle
C
ELLIPTIC ELLIPTIC GEOMETRYGEOMETRYBernard Riemman
Spherical/Earth ModelElliptic Points: All points ON SURFACE of sphereElliptic Lines: Great circles (diameter = equator)Antipodal point
ELLIPTIC ELLIPTIC GEOMETRYGEOMETRYBernard Riemman
Spherical/Earth ModelElliptic Points: All points ON SURFACE of sphereElliptic Lines: Great circles (diameter = equator)Antipodal point
COMPARISONCOMPARISONEuclidean Hyperbolic Elliptic
Any 2 lines intersect in
1 point 1 point 1 point/s
Parallel lines are equidistant Converge in 1 direction; diverge in the other
DNE
If a line intersect 1 of 2 parallel lines, it must
Intersect the other
May or May not intersect the other
NA
2 lines perpendicular to a line are
parallel parallel Intersecting
Sum of angles in a triangle
180 < 180 > 180
Area of triangle Independent of sum of the angles
Proportional to the deficit of sum
Proportional to the excess of sum
2 triangles with corresponding angles congruent
similar congruent congruent
PROJECTIVE PROJECTIVE
GEOMETRYGEOMETRY Johannes KeplerVictor PonceletPrinciple of Perspectivity: applied on a 2D canvas
to depict 3DA 1-1 correspondence Ideal points, ideal line, ideal plane
PROJECTIVE GEOMETRYPROJECTIVE GEOMETRYPrinciple of Perspectivity: applied on a 2D canvas
to depict 3D
TOPOLOGYTOPOLOGYFigures (mathematical spaces) whose geometric
properties are unchanged by continuous deformation (stretching, twisting, shrinking)
Topological transformation : continuous deformation
Properties preserved: outside/insideNot preserved: shape, magnitudeTopologically equivalent: objects changed into
another by a topological deformation
EUCLIDEAN GEOMETRYEUCLIDEAN GEOMETRYI. Triangle Properties a) sum of any 2 sides of a triangle is greater than the 3rd
side 2,3,6 2,3,5 2,3,4 b) sum of measures of the angle of a triangle is 180
c) congruence: same shape, same measurement SAS ASA SSS d) similarity: same shape e) area and perimeter rectangle: A = l x w P = 2l + 2w triangle: A = (1/2)bh P = add all 3 sides circle: A = 𝜋r2 C = 2𝜋r
TRIGONOMETRYTRIGONOMETRYRight triangleSOHCAHTOAsine A = sin A = side opposite/hypothenusecosine A = cos A = side adjacent/hypotenuse tangent A = tan A = sin A/ cos A = side opposite/
side adjacentcosecant A = csc A = 1/sin Asecant A = sec A = 1/cos Acotangent A = cot A = 1/tan A = cos A /sin AAngle of elevation/depression
TRIGONOMETRYTRIGONOMETRYAn airplane 405 ft above a landing field when the
pilot cuts out his motor. He glides to a landing at an angle of 130 with the field. How far will he glide in reaching the field ?
A road running from the bottom of the hill to the top is 625 m. If the hill is 54 m high, what is the angle of elevation of the road ?