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MATH 205 [email protected] This Form Is a Review Sheet for MATH 090 Students Based Primarily Around the Teaching & Instructions of Professor V

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Created by: Jasmine E. Aue MATH 205 [email protected] This form is a review sheet for MATH 090 students based primarily around the teaching & instructions of Professor V. Smith of ESU. HELPFUL TIPS FOR COURSE CONTENT (​ie. not formulas but important​) FORMULA LIST ℉ = ((℃/5)(9)) + 32 | Lines, Arrows, Points & Planes Fahrenheit to Celsius formula This section is a review of lines, arrows, points & planes. How they are defined, look and interact with each other. ℃ = ((℉ − 32)/9)(5) ​ |​ Celsius to ❏ Collinear​: same line Fahrenheit formula ❏ 2 Coplanar​: same plane a2 + b = c2 |​ Pythagorean ❏ ● ● Closed​: a line with an endpoint on each end, kind of like this Theorem ❏ Half​: a line with only one endpoint, kind of like this ● d = (x − x ) + (y − y ) ❏ Open​: a line with no endpoints, like this √ 2 1 2 1 | ❏ → ⋃ → = ↔ Diameter of a Circle ❏ → ⋂ → = r2 = (x − h)2 + (y − k)2 |​ Radius ❏ ●● ⋃ ●→ = → Formula ❏ FE￮ ⋂ ￮ ￮ ￮ → → = a2 + b2 = c2 |​ Pythagorean ❏ Intersecting Lines​: lines that have only one point in common ❏ Intersecting Planes​: planes that have only one line in common Theorem ❏ Skew​: cannot be coplanar y = mx + b ​ |​ y is from (x,y), m ❏ Concurrent​: 3 or more intersections at the same point is the slope, x is from (x,y) & b ❏ Parallel​: same slope, non-intersecting is the y-intercept m = (y2 − y1) / (x2 − x1) |​ this is ❏ There is exactly one line containing any two distinct points the point-slope formula, use ❏ If two points are in a plane then the line of points are in a plane this to find the slope of a line ❏ If two planes intersect, there is a line at that intersection given two coordinate pairs ❏ There is exactly one plane that contains any 3 distinct (x1, y1) & (x2, y2) noncollinear points P = 4s ​ |​ Perimeter of a Square ❏ Collinear points/lines can have infinitely many planes 2 ❏ One line + one point = a unique line A = s ​ |​ Area of a Square ❏ Parallel lines are on one plane P = 2w + 2l ​ |​ Perimeter of a ❏ Two intersecting lines are on one plane Rectangle A = lw ​ |​ Area of a rectangle C = πd = 2πr ​ |​ Circumference of a Circle A = πr2 ​ |​ Area of a Circle P = a + b + c ​ |​ Perimeter of a Triangle A = 1 bh ​ |​ Area of a Triangle ABAY 2 Angles & Circles A = bh ​ |​ Area of a This section is a review of angles and circles, not including their formulas (​see formula list for formulas​). Parallelogram ❏ Complementary Angles: add to 90° 1 A = 2 (b1 + b2)h ​ |​ Area of a ❏ Supplementary Angles: add to 180° Trapezoid ❏ Acute Angles: less than 90° A = 1 d d ​ |​ Area of a Diamond ❏ Right Angles: equal to 90° 2 1 2 ❏ Obtuse Angles: greater than 90° but less than 180° SA = 2B + P h ​ |​ Surface Area of ❏ Reflex Angles: greater than 180° but less than 360° a Right Prism, where B = area ❏ Straight Angles: equal to 180° of the base and P =perimeter ❏ Circle: 360° of the base ❏ 1 360 = 1° = 60′ = 3600" (when going from degrees to minutes to V = Bh ​ |​ Volume of a Right seconds, multiply & when going from seconds to minutes to Prism degrees, divide) SA = 2lh + 2wh + 2lw ​ |​ Surface Diagrams Related to Angles and Circles: l Area of a Rectangular Prism V = lwh ​ |​ Volume of a Rectangular Prism 2 SA = 6s ​ |​ Surface Area of a me Cube V = s3 ​ |​ Volume of a Cube SA = 2πrh + 2πr2 ​ |​ Surface Area of a Right Cylinder (regular cylinder like a pringles can) N V = πr2h ​ |​ Volume of a Right Cylinder 2 SA = 4πr ​ |​Surface Area of a Interior Angles: 3, 4, 5, 6 Sphere Exterior Angles: 1, 2, 7, 8 4 2 V = 3 πr ​ |​ Volume of a Sphere Alternate Interior Angles: 3 & 6, 4 & 5 SA = B + 1 P l ​ |​ Surface Area of Alternate Exterior Angles: 1 & 8, 2 & 7 2 Corresponding Angles: 1 & 5, 2 & 6, 3 & 7, 4 & 8 a Pyramid (using slant height) lwh V = 3 ​ |​Volume of a Pyramid WHN Arc Minor 2 SA = πr√r2 + h + πr2 ​ |​ Surface I the transversal 2 2 l is Chord Area of a Cone where √r + h is the slant height \ 1 2 V = 3 πr h ​ |​ Volume of a Cone r Center Arc Major K(O,r) Radius d = a + b Circle “name” Normal Distribution This is a diagram of Normal Distribution: STUDY TIPS Scheduling |​ keep up to date with all assignments, quizzes, tests, exams, etc. by documenting them onto a calendar you regularly check, using a reminders app, a planner or other scheduling tool(s) Review |​ attend regular tutoring sessions, create your own mock tests using practice Shapes problems, review past HW, quizzes, tests & exams This section is a review of shapes, triangles will have their own more detailed section following this one. (​see formula list for formulas on area, perimeter, volume, etc.​) Eat Well |​maintain a healthy, ❏ Simple​: does NOT intersect itself balanced diet to increase ❏ Closed​: the start of the shape is also its end & vice versa. brain function (memory, ❏ Regular Polygon​: equilateral (sides) & equiangular (angles) engagement, attention, etc.) ❏ Trapezoid​: at least one pair of parallel lines & to decrease fatigue, ❏ Kite​: 2 pairs of adjacent & congruent angles fogginess, etc. ❏ Parallelogram​: Both pairs of opposite sides are parallel ❏ Rhombus​: two adjacent sides are congruent Sleep Well |​ maintain a health sleep schedule of 6-8 hours of Diagrams Related to Shapes: sleep per night TUTORING INFORMATION .. ... ESU Tutoring Website | https://www.esu.edu/tutoring g. /index.cfm ESU’s WCOnline Tutoring Link | https://esu.mywconline.com " - " / i I wearE T Ex . every quay is a rectangle . every trapezoid is a quadrilateral . NOT is a . EX . every rectangle quay Not every rhomboid is a rectangle . Triangles This section is a review of Triangles. (​see formula list for formulas for area, perimeter, etc.​) Diagrams Related to Triangles: Special Triangles Miscellaneous Information This section is a review of miscellaneous information that does not fit a big enough category to be a separate section. These concepts are just as important though! ❏ Area is always squared (x2) ❏ Volume is always cubed (x3) π ❏ When going from x° to radians; x° • 180° 180° ❏ When going from radians to x° ; x • π ❏ Kilo (1000), Hecto (100), Deka (10), GRAM (Base Measurement, 1),Deci (0.1), Centi (0.01), Milli (0.001) ❏ 1 Ton = 1000000g = 1000kg ❏ 1kg = 1000g ❏ 1hg = 100g ❏ 1dkg = 10g ❏ 10dg = 1g ❏ 100cg = 1g ❏ 1000mg = 1g ❏ 1cm3 = 1mL = 1g ❏ 1dm3 = 1L = 1kg ❏ 1yd2 = 36in2 ❏ 1ft = 12in ❏ 1yd = 3ft = 36in ❏ 1mi = 1760yd = 5280ft = 63360in Vectors: iBBq Name___________________ GEOMETRY FINAL EXAM REVIEW I. MATCHING _____reflexive A. a(b + c) = ab + ac _____transitive B. If a = b & b = c, then a = c. _____symmetric C. If D lies between A and B, then AD + DB = AB. _____substitution D. If a = b, then b = a. _____distributive E. a = a _____definition of midpoint F. If D is the midpoint of ̅̅̅ ̅, then AD = AB. _____midpoint theorem G. If a + b = c and a = d, then d + b = c. _____segment addition postulate H. If D is the midpoint of ̅̅̅ ̅, then AD = DB. II. Fill in the blank. 1. An equilateral triangle is also a(n) ___________triangle. 2. The ____________ is the longest side of a right triangle. 3. Similar triangles have congruent corresponding ____________ and the corresponding __________ are in proportion. 4. In an isosceles triangle, the __________ angle is the angle that is different. 5. The __________ of a triangle is a segment from a vertex to the midpoint of the opposite side. 6. A(n) ____________ of a triangle is a segment from a vertex ^ to the opposite side. 7. A(n) ____________ ____________ of a segment is a line, segment, or ray ^ to a segment at its midpoint. 8. The measure of a central angle is _________ to its intercepted arc. 9. Two __________________ angles have a sum of 90è. 10. Two __________________ angles have a sum of 180è. 11. A __________ has only 1 endpoint. 12. If two lines are ____________, they form right angles. 13. Two lines intersect in a ____________. 14. Two planes intersect in a ____________. 15. Through any three collinear points there are ____________ ___________. Through any three non-collinear points there is __________ ________ ________. 16. __________ angles measure between 0è and 90è. 17. __________ angles measure between 90è and 180è. 18. Find the side of square with area 16 units2. ____________. 19. If the ratio of the measures of the angles of a triangle is 2:2:5, then the triangle is a(n) ____________ triangle. 20. If 4 points all lie on the same line, then the points are ____________. 21. The interior angle sum of a hexagon is ____________. 22. The exterior angle sum of a decagon is ____________. 23. If each interior angle of a regular polygon is 144, then the polygon is a __________. 24. If each exterior angle of a regular polygon is 30, then the polygon has ____________ sides. 25. In a 30è - 60è - 90è triangle, the long leg is ______ times the short leg. 26. In a 45è - 45è - 90è triangle, the hypotenuse is ______ times the leg. 27. An angle inscribed in a semicircle is a __________ angle. 28. Write 32 in simplest radical form. ______ 29. If Ø A is a right angle and m Ø A = (4x + 10)°, then x = __________. 1 2 a 3 4 a || b 5 6 b 7 8 30. Ø 3 & Ø 5 are ____________ angles & therefore are ____________. 31. Ø 4 & Ø 5 are ____________ angles & therefore are ____________. 32. Ø 2 & Ø 6 are ____________ angles & therefore are ____________. 33. If m Ø 6 = (x + 5)° and m Ø 4 = (2x + 10)°, then m Ø 4 =______.
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  • Six Mathematical Gems from the History of Distance Geometry

    Six Mathematical Gems from the History of Distance Geometry

    Six mathematical gems from the history of Distance Geometry Leo Liberti1, Carlile Lavor2 1 CNRS LIX, Ecole´ Polytechnique, F-91128 Palaiseau, France Email:[email protected] 2 IMECC, University of Campinas, 13081-970, Campinas-SP, Brazil Email:[email protected] February 28, 2015 Abstract This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron’s formula, Cauchy’s theorem on the rigidity of polyhedra, Cayley’s generalization of Heron’s formula to higher dimensions, Menger’s characterization of abstract semi-metric spaces, a result of G¨odel on metric spaces on the sphere, and Schoenberg’s equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling. Keywords: Euler’s conjecture, Cayley-Menger determinants, Multidimensional scaling, Euclidean Distance Matrix 1 Introduction Distance Geometry (DG) is the study of geometry with the basic entity being distance (instead of lines, planes, circles, polyhedra, conics, surfaces and varieties). As did much of Mathematics, it all began with the Greeks: specifically Heron, or Hero, of Alexandria, sometime between 150BC and 250AD, who showed how to compute the area of a triangle given its side lengths [36]. After a hiatus of almost two thousand years, we reach Arthur Cayley’s: the first paper of volume I of his Collected Papers, dated 1841, is about the relationships between the distances of five points in space [7]. The gist of what he showed is that a tetrahedron can only exist in a plane if it is flat (in fact, he discussed the situation in one more dimension).