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MODULE 06: S hapes ( Pt. 1) Geometry was developed as a practical way of dealing with areas, lengths, and volumes. - geo - earth - metron - measurement Geometry before Euclid 1. Babylonian civilization - clay tablets with cuneiform script 3. Eratosthenes - first mathematician to calculate the circumference of the earth using a tower’s shadow, the distance of Alexandria and Syene, - calculated value: 250,000 stadia or around 40,000 km - actual value: 40,075 km - VIDEO: http://bit.do/shadowcircum 4. Pythagoras - Pythagorean theorem a2 + b2 = c2 - - wrote formulas for finding areas and volumes of shapes like circles and cylinders approximated π π to 3 circumference = 3 3 • diameter d iameter 1 area = 12 circumference2 had the concept of Pythagorean P ythagorean triples but no sources indicate they know the Pythagorean theorem 2. Egyptian civilization - flourished almost at the same time as the Babylonian civilization - formulas in finding areas and volumes were essential in the construction of famous pyramids pyramids and the determination of food supply; recorded on the Ahmes A hmes Papyrus a ka Rhind R hind Mathematical Papyrus 5. Archimedes of Syracuse - volume of an irregularly shaped object = volume displaced when submerged in water - derived an accurate approximation of the value of π using the method method of exhaustion developed by Eudoxus of Cnidus 6. Greeks - irrational numbers e.g. √2 , π , golden ratio ( ϕ ) *refer to previous trans (04 Module 5) for explanation of golden ratio; alam ko namang naprint/nasave niyo na yun : ^) NOTE: ϕ = 1.618… 1 ϕ = 0.618... Euclidean geometry - by mathematician Euclid E uclid of Alexandria - first to show how results stated by earlier mathematicians fit into a comprehensive deductive and logical mathematical system - The Elements - Euclid’s book; collection of postulates, definitions, proofs, etc - oldest geometry conceived; geometry studied currently - undefined terms in geometry: p oints, lines, planes Euclidean axioms formulated to develop a structured mathematical system 1. Things that are equal to the same thing are equal. 2. If equals are added to equals, then the whole are equal. 3. If equals are subtracted from equal, the remainders are equal. 4. Things that coincide with one another are equal to one another. 5. The whole is greater than the part. Euclidean postulates govern Euclid’s axiomatic system 1. A straight line can be drawn from any point to any point. - point 1, point 2, connect sila → line 2. A finite straight line can be produced continuously in a straight line. - maraming line segment sa loob ng straight line 3. A circle may be drawn with any point as center and any distance as radius. - stationary point 1, point 2 somewhere then ikutan yung point 1 (equal distance/radius) → charan circle 4. All right angles are equal to one another. - kahit anong paikot sa mundo o sayo, 90° = 90° 5. Called parallel postulate. If a transversal falls on two lines in such a way that the interior angle on one side of the transversal are less than two right angles, then the lines will intersect on that side on which the angles are less than two right angles. - if interior angles on one side of the transversal line are both <90°, °, the two lines will intersect when extended towards the said side - dito galing ang if the sum of the interior angles are 180°°, the two lines are parallel - “parallel postulate” pero hindi directly tungkol sa parallelism? enter Playfair’s axiom Playfair’s axiom (based on 5th postulate) Through a point P not on line l , there exists exactly one line ( m ) passing through point P parallel to l . Lines s and q are not parallel to l . Euclidean triangles - sum of the measures of interior angles is always 180° Congruence (≡) criteria 1. SSS (side - side - side) 3 corresponding sides are congruent. Congruent vs similar Euclidean triangles Congruent Similar same shape and size same shape but not necessarily same size equal corresponding angles and sides equal corresponding angles, proportional corresponding sides NOTE: All congruent triangles are similar. Not all similar triangles are congruent. △ABC ≡ △XY Z 2. SAS (side - angle - side) 2 corresponding sides and the angle angle between them are congruent. Consider the 2 congruent triangles below. ✓ same shape ✓ same size - corresponding sides are of the same lengths AB = X Y BC = Y Z AC = X Z - corresponding angles are equal m∠A = m∠X △ABC ≡ △XY Z m∠B = m∠Y m∠C = m∠Z 3. ASA (angle - side - angle) 2 corresponding angles and the side side between them are congruent. △ABC ≡ △XY Z 4. AAS* (angle - angle - side) 2 corresponding angles and a non-included non-included side are congruent. Non-Euclidean geometry Hyperbolic geometry Poincaré Disk △ABC ≡ △XY Z *hindi kasama sa ppt ni sir pero kasama sa internet Similarity (~) criteria *hindi rin kasama sa discussion/ppt ni sir pero for reference lang kung gusto icompare sa congruence Through a point P not on a line l, there exists at least two lines passing through point P parallel to line l. - sum of hyperbolic triangle’s interior angles is less than 180° - similar hyperbolic triangles are congruent Elliptic geometry 1. SSS (side - side - side) 3 corresponding sides are in proportion. AB BC XZ XY = Y Z = AC ABC ~ X Y Z 2. SAS (side - angle - side) 2 corresponding sides are proportional, and the angle angle between them are congruent. AB BC XY = Y Z m∠B = m∠Y ABC ~ X Y Z 3. AA/AAA (internal screams??) 2 or 3 corresponding angles are congruent. m∠A = m∠X m∠B = m∠Y m∠C = m∠Z Through a point P not on a line l, there exists no line passing through point P parallel to line l. - sum of elliptical triangle’s interior angles is more than 180° - similar elliptical triangles are congruent Application: in sea and air navigation, spherical model of elliptic geometry is used in calculating distances and planning flight paths Projective geometry - includes mathematical s tudy of vanishing points and perspective - three-dimensional (3D) visualization - artists’ portrayal of humans in creating works of art (Renaissance period) - 3D images in games (modern times) Gerard Desargues - father of projective geometry - stated that any two lines on a plane intersect giving rise to creation of projective geometry - revisited Euclid’s Euclid’s second postulate and considered the concept of parallel lines interesting in a plane at an ideal point Two triangles are perspective from a point P if sides Aa, Bb and Cc intersect at point p. The point of intersection is called the center of perspectivity. Two triangles ABC and abc are perspective perspective from line l if corresponding sides of the triangles meet at points on l, called the axis a xis of perspectivity. Desargues Theorem If two triangles are perspective from a point, then they are perspective from a line. Geometries described previously (Euclidean and non-Euclidean) talked about geometries with very v ery concrete notions of space. Space was conceptualized as a set of points t ogether with an abstract abstract set of relations i n which these points are involved. Topology - an invention of many mathematicians (e.g. Cantor, Euler, Frechet, Huasdorf, Mobius and Riemann) - focuses on discovering and analyzing the essential similarities and differences between sets and figures Topological transformations - stretching, shrinking, twisting or bending the figure in any way that allows the points to remain distinct Two figures are called topologically t opologically equivalent if one figure can be stretched, shrunk, twisted or bent in to the same shape as the other A square, a circle, and a triangle are topologically equivalent. The coffee and the cup are topologically equivalent, but a coffee cup and a muffin are not.