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Synthetic Geometry

Synthetic geometry differs from other branches of geometry because it focuses on pure geometrical contents and draws conclusions through the use of axioms, logical arguments and theorems. Synthetic geometry analyzes the incidence relationship between objects such as lines and planes. These objects and their relationship of incidence are known as the primitives of synthetic geometry.

Axiomatic geometry is sometimes used interchangeably with the term synthetic geometry. In mathematics, the concept of an axiom is critical to the study of Euclidean geometry. Axioms are statements which are used to describe and prove the primitives of synthetic geometry. Essentially, these are just basic statements which have not yet been accepted or proven. For example, the statement “for any two distinct points, there is only one line which passes through both of these points”, is an axiom.

In conjunction with axioms, theorems are used in synthetic geometry. Theorems differ from axioms because they represent proven statements. Theorems can be proven by axioms which are thought to be accepted or by other theorems which exist. In order to formulate conclusions regarding mathematical problems and conduct geometric computations, theorems are utilized in synthetic geometry. Furthermore, logical arguments are used as a mode of proof through persuasion.

Evidently, synthetic geometry is one of the many distinct branches of geometry, just like analytic and algebraic geometry, which is used to understand geometrical structures. Thus, understanding the proper use of axioms, logical arguments and theorems is essential to this field of thought. 

Uses of the Pythagorean Theorem

Geometry is a very broad field of mathematics composed of a wide range of tools that can be used for problem solving. In this module, you are going to research three examples of the implementation of geometry that would employ the use of the Pythagorean Theorem as a problem-solving tool. The examples you find can come from s

We wish to manufacture a cylindrical can of height h and radius r, given in inches. The ends cost 10 cents per square inch and the sides cost 5 cents per square inch to manufacture. If the desired volume of the can is 24 in3, what is the total cost of manufacturing the can? (See the attached question file.)

We wish to manufacture a cylindrical can of height h and radius r, given in inches. The ends cost 10 cents per square inch and the sides cost 5 cents per square inch to manufacture. If the desired volume of the can is 24 in3, what is the total cost of manufacturing the can? (See the attached question file.)

An army cadet is involved in a compass exercise.

An army cadet is involved in a compass exercise. He leaves point O and walks 50m due west to point A. He then walks 80m due north to point B and finally 60m on a bearing of 320 deg to point C. a) Illustrate this information on a sketch b) Calculate the distance and bearing of B from O c) Calculate the distance and bearing

Rearranging Equations

Volume of right circular cone is V=1/3 (pi)r^2h if height is twice the radius express volume V as as a function of r.

Critical numbers

(1) Given function: f(x) = (x^2)*[e^(12x)] (a) Find its horizontal asymptotes. (b) Find its critical numbers. (c) Find its inflection points. (2) Given function: f(x) = (x^3) / [(x^2) - 16] (a) Find its critical numbers. (b) Find its inflection points.

Path connected problems

(See attached file for full problem description with proper symbols) --- ? Show that, for , the sphere is path connected. ? Show that if f:X->Y is a continuous map between topological spaces and X is path connected, then the image f(Y) is also path connected. ---

Euclidean Metric : Translation and Rotation

Let be the standard euclidean metric on defined by where and are any points in . a) a translation is a map given by for some fixed given point . Prove, that the Euclidean metric d on is translation-invariant, ie, for any translation T it follows: . b) A rotation is a map given by

1. Let α:={...}. Is this an open cover for (0,2)? Can you find a subcover for α for (0,2)? 2. Find an open cover for (0,2) which consists of open balls of the form B1(z) with radius 1, which does not contain any finite subcover.

1. Let α:={...}. Is this an open cover for (0,2)? Can you find a subcover for α for (0,2)? 2. Find an open cover for (0,2) which consists of open balls of the form B1(z) with radius 1, which does not contain any finite subcover. Please see attached file for full problem description.

Subset Game involving Intervals and Subintervals

In the following infinite game, Alice and John take turns moving. First, Alice picks a closed interval I<sub>1</sub> of length <1. Then, Bob picks a closed subinterval I<sub>2</sub> which is a subset of or equal to I<sub>1</sub>, of length 1/2. Next, Alice picks a closed subinterval I<sub>3</sub> which is a subset of or equal t

Regular Solids

For each of the 5 regular polyhedra, enumerate the number of vertices (v), edges (e), and faces (f), and then evaluate the quantity v - e + f. (One of the most interesting theorems relating to any convex polyhedron is that v - e + f = 2. -> This was given as part of the problem. I am not sure if it is of any value when solving t

Convergence of sequence in Rn

Please see the attached file for full problem description. Let A and B be closed subset of Rn with A &#8745; B = Ø. a. Prove that &#8704;u &#8712; A, &#8707;_ > 0 such that N_ (u) &#8745; B = Ø b. Prove that there is an open set OA satisfying OA &#8835; A and OA &#8745; B = Ø c. Prove or find a counterexa

Identifying types of conics

For each of the equations identify the type of conic it is and list the major features associated with this type of conic Major Features Circles - Find the coordinates of the center - Find the length of the radius - Find the length of the diameter Parabolas - Find the coordinates of the vertex - Find the coordinates

Inequality

Suppose that there are three points A,B and C on a line where C is between A and B. o is any other point not on the line. Prove that OA+OB>OC