CHAPTER I
INTRODUCTION
Man’s capability to reason out was the most crucial factor in his existence. His ability to give meanings to the things perceive by the eyes was natural for him. Thus, his concern about spaces increases. Based on man’s experiences, spaces are simply related to dimensions and figures and can be interpreted using mathematical expressions. Through this, the two disciplines called topology and geometry come in significance. In mathematics, topology is a branch concerned with the study of topological spaces. (The term topology is also used for a system of open sets used to define topological spaces, but this article focuses on the branch of mathematics. Wiring and computer network topologies are discussed in network topology.) Roughly speaking, topology is the study of geometric objects without considering their dimensions. Topology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study. On the other hand, geometry was defined as the branch of mathematics that treat space and its relation, especially as shown in the properties and measurement of points, lines, angles, surfaces and solids.
No one will ever be able to prove that the line [i.e. the perimeter] can be equal to the area [of the triangle in general]: as there may be different sizes, among which nothing can be done either with being equal or a ratio. But yet, if it has been attained, the whole length measured to comprise the perimeter as many times as the area consists of the square of the same; in a way we will be able to say, this area to be equal to the same perimeter.
Background of the Study
This study entitled Conditions on Triangles having equal areas and perimeters is an exposition of generating and finding methods on when the area of a triangle numerically equal to its perimeter.
Several studies have been conducted on triangles having areas and perimeters of equal numerical value. Markowitz (1981) determined that only five triangles having sides with integral lengths exist for which the areas equal the perimeters, but that infinitely many have sides of rational lengths. Markowitz (1981) and Bates (1979) both determine equations satisfied by triangles having equal areas and perimeters, but these equations can only be used through trial and error.
The studies of these authors suggest new idea on investigating triangles of equal area and perimeter and shows different approaches and methods of each other that one is urged to further study the concept of the subject. Inspired by the ideas presented by these authors, this current study has investigated the geometric characteristics of such triangle and extracts theorems on the conditions of triangles having areas and perimeters numerically equal.
Statement of the Problem
The purpose of this study is to present conditions on when the triangles have areas and perimeters numerically equal. Line segments, tangent lines, angles, triangles and inscribed circles are used to construct representation on deriving triangles of equal area and perimeter.
Specifically, it attempted to formulate theorems on;
1. when the triangles have areas and perimeters numerically equal
2. when the triangles have areas and perimeters equal in terms of the length of sides
3. when the triangles have areas and perimeters equal in terms of the length of tangent segments
4. when the triangles have areas and perimeters equal in terms of the deviation of x from the midpoint of the tangent segments
5. when the triangles have areas and perimeters equal in terms of the radius xy/2 where x and y are distance from the midpoint connected to two tangent lines having a common point
Significance of the Study This study is significant to individuals who are fun of investigating and finding proofs on geometric construction specifically, the geometric construction that involves lines, triangles and inscribed circle. This will help them improve their comprehension skills and logical thinking especially on pure mental construction called “concept”. Objectively, the result of this study is beneficial to teachers and students in the field of mathematics teaching-learning process because this expresses idea in accordance with the categories of mathematics learning objects.
Hence, in response to a great pressure for students to succeed in mathematics, this research paper is important as;
1. a resource material to mathematically inclined individuals to the area of algebra, geometry, and number theory,
2. a guide for them to develop high reasoning power and proving theorems,
3. a source material for enrichment for the mathematically gifted students,
4. moreover, this study is important to some related problems involving line segments, tangent lines, triangles and inscribed circle.
Furthermore, this study encourages the readers to construct another geometric representation to extend the concept of this study.
Scope and Delimitation of the Study
Many mathematical ideas become clear by looking at patterns. Our intuition can lead to conjectures, theorems, and often new proofs. Such in the case of studying the relation between the area of a triangle and its perimeter. Because a triangle with a given perimeter can yield several different areas, my study will be limited to triangles with integral sides greater than 4. The construction of circle of radius 2 inscribed by a triangle of general form (Equilateral, Isosceles, Scalene) were presented in a basic Euclid’s concept to find solutions for this proposed study. Premises and conjectures were fully explained and successfully proven, that leads to a generalization in the form of theorems. Basic concept on inscribed circle, tangent segments, areas and perimeters as well as other geometric methods were included to facilitate proofs of the theorems
Definition of Terms
To provide clarity of the terms used in this study, the following Definitions are clearly defined:
Triangle – is a polygon formed by the union of three non- collinear segments.
Circle – is a set of points that are equidistant from a fixed point called the center of the circle.
Radius of a circle – is a segment with endpoints at the center of the circle and a point on the circle.
Tangent Segment – is a segment that lies on a line tangent to a circle with one of its endpoints at the point of tangency.
Area of a Triangle – is one-half the product of its base and altitude.
Perimeter of a Triangle – is the sum of the lengths of the sides of the triangle.
Credit:ivythesis.typepad.com
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