Geometric figures

Armenian state pedagogical university after khachatur abovyan

Faculty of Mathematics, Physics and Informatics

Topic: Geometric figures

Piruza Mkhitaryan

Scientific Supervisor:

K. Baghdasaryan

Yerevan 2013

Table of content

Introduction

1. Polygons

2. Triangles

3. Quadrilateral

4. Circles

Conclusion

References

Glossary

Introduction

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.

There are basic geometric figures that are accepted without definition and are used for defining other concepts, such as point, line, plan and space concepts. In addition to this there are geometric objects, which are widely used in mathematics, physics and other fields. Geometric objects can be two-dimensional, three-dimensional and n-dimensional. We’ll explore two-dimensional objects.

Twodimensional objects are called geometric figures, which are located on the plane. Examples of the geometric images are triangles, quadrilateral, polygons, circle, etc.

Three-dimensional objects are located in the space. Examples of three-dimensional objects are the cube, tetrahedron, prism, cone, cylinder, sphere, etc.

Twodimensional and three-dimensional objects also are called geometric shapes.

In this study I want to introduce some properties of two-dimensional geometric shapes or geometric figures and interesting facts about them.

1. Polygons

A polygon can be defined as a geometric object «consisting of a number of points (called vertices) and an equal number of line segments (called sides). In other words, a polygon is closed broken line lying in a plane»

A polygon with vertices (and sides) is known as angon. A polygon for which the only points of the plane belonging to two polygon edges of are the polygon vertices is said to be a simple polygon.

If all sides and angles are equivalent, the polygon is called regular. Polygons can be convex, concave, or star. The following table gives the names for polygons with sides. The words for polygons with sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, depending on context. It is therefore always best to specify «regulargon» explicitly. For some polygons, several different terms are used interchangeably, e.g., nonagon and enneagon both refer to the polygon with sides.

2. Triangles

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.

The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, «It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?»).

It is common to label the vertices of a triangle in counterclockwise order as either, (or). The vertex angles are then given the same symbols as the vertices themselves. The symbols (or) are also sometimes used, but this convention results in unnecessary confusion with the common notation for trilinear coordinates, and so is not recommended. The sides opposite the angles (are then labeled with these symbols also indicating the lengths of the sides.

There are different types of triangles.

Triangles can be classified according to the relative lengths of their sides:

· In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.

· In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles.

· In a scalene triangle, all sides are unequal, and equivalently all angles are unequal.

In diagrams representing triangles (and other geometric figures), «tick» marks along the sides are used to denote sides of equal lengths — the equilateral triangle has tick marks on all 3 sides, the isosceles on 2 sides. The scalene has single, double, and triple tick marks, indicating that no sides are equal. Similarly, arcs on the inside of the vertices are used to indicate equal angles. The equilateral triangle indicates all 3 angles are equal; the isosceles shows 2 identical angles. The scalene indicates by 1, 2, and 3 arcs that no angles are equal.

Triangles can also be classified according to their internal angles, measured here in degrees.

· A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs or of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3−4-5 right triangle, where 3² + 4² = 5². In this situation, 3, 4, and 5 are a Pythagorean Triple. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees.

There is interesting fact about right triangle: for every two natural numbers numbers are sides of right triangle.

· Triangles that do not have an angle that measures 90° are called oblique triangles.

· A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.

· A triangle that has one interior angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle.

A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral.

3. Quadrilateral

A quadrilateral is a polygon with four sides (or edges) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. polygon triangle quadrilateral circle

There are three topological types of quadrilaterals convex quadrilaterals (left figure), concave quadrilaterals (middle figure), and crossed quadrilaterals (or butterflies, or bow-ties; right figure).

In addition to this there are many types of convex quadrilaterals, such as a parallelogram, rectangle, rhombus, square, trapezoid and kite.

A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles «a» are the same, and angles «b» are the same).

A rectangle is a four-sided shape where every angle is a right angle (90°). Also opposite sides are parallel and of equal length.

A rhombus is a four-sided shape where all sides have equal length. Also opposite sides are parallel and opposite angles are equal. Another interesting thing is that the diagonals meet in the middle at a right angle. In other words they «bisect» (cut in half) each other at right angles.

A square has equal sides and every angle is a right angle (90°). Also opposite sides are parallel.

A trapezoid has a pair of opposite sides parallel.

A kite has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.

4. Circles

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. The distance from the center is called the radius, and the point is called the center. Twice the radius is known as the diameter. The angle a circle subtends from its center is a full angle, equal to or radians.

A line that goes from one point to another on the circle’s circumference is called a chord. A line that «just touches» the circle as it passes by is called a tangent. And a part of the circumference is called an arc.

There are two main «slices» of a circle

The «pizza» slice is called a sector.

And the slice made by a chord is called a segment.

Conclusion

In this paper I have tried to introduce several two-dimensional geometric figures and some of their properties. I introduced triangles, quadrilaterals, polygons, their classifying by different bases, circle and its parts, which are used for solving practical problems.

In school the course of geometry consist of two parts — plan geometry and space geometry. In school courses plane geometry or two-dimensional geometric figures are studied in 7th, 8th and 9th grade and they are the basis of geometry. Space geometry or three-dimensional geometric shapes are studied in 10th, 11th and 12th grade, but during the study of three-dimensional geometric shapes two-dimensional geometric figures are widely used. In addition to this two-dimensional geometric figures are defined as parts of three-dimensional and n-dimensional shapes. Our world is threedimensional, that’s why we must know common properties of three-dimensional geometric shapes, therefore we must recognize also two-dimensional geometric figures.

Two-dimensional geometric figures are widely used in the practise, so everybody must know some properties of geometric figures and their applications.

Finally geometry is very beautiful and practice subject and it develops the student’s analytical and visual thinking, which explains its essential role in our life.

Glossary

1. Pointpoints are zero-dimensional surface

2. Linea line is an infinite set of points of the form

3. Plana plane is a flat, two-dimensional surface

4. Geometric shapesTwodimensional and three-dimensional objects are called geometric shapes

5. PolygonA polygon is a geometric object consisting of a number of points and an equal number of line segments namely a cyclically ordered set of points in a plane

6. -gonA polygon with vertices and sides is known as angon

7. Convex polygon-A planar polygon is convex if it contains all the line segments connecting any pair of its points

8. Concave polygon-A concave polygon is a polygon that is not convex

9. TriangleA triangle is a 3-sided polygon sometimes called the trigon

10. Triangle geometryThe study of triangles is sometimes known as triangle geometry

11. Equilateral triangle — The triangle where all sides have the same length

12. Isosceles triangle-the triangle where two sides are equal in length

13. Scalene trianglethe triangle where all sides and all angles are unequal

14. Right triangle or right-angle trianglethe triangle that has a right interior angle

15. HypotenuseThe side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle

16. LegsThe other two sides of right triangle are called the legs

17. Oblique trianglesTriangles that do not have an angle that measures 90°

18. Acute triangle or acute-angled triangle.- A triangle that has all interior angles measuring less than 90°

19. Obtuse triangle or obtuse-angled triangleA triangle that has one interior angle that measures more than 90°

20. Quadrilateral — A quadrilateral is a polygon with four sides (or edges) and four vertices or corners

21. Parallelograma quadrilateral, which opposite sides are parallel and equal in length

22. Rectanglea quadrilateral, which has four right angles (90°). Also opposite sides are parallel and of equal length.

23. RhombusA quadrilateral, where all sides have equal length

24. Square-A quadrilateral, which has equal sides and every angle is a right angle (90°).

25. Trapezoida quadrilateral, which has a pair of opposite sides parallel

26. Kitea quadrilateral, which has two pair sides, Each pair is made up of adjacent sides that are equal in length

27. Circle-is two-dimensional shapes that is the set of all points in a plane that are a given distance from a given point

28. CenterA point inside the circle. All points on the circle are equidistant (same distance) from the center point

29. Radius-The radius is the distance from the center to any point on the circle. It is half the diameter.

30. DiameterThe distance across the circle. The length of any chord passing through the center. It is twice the radius

31. Circumferencethe circumference is the distance around the circle

32. ChordA line segment linking any two points on a circle

33. TangentA line passing a circle and touching it at just one point

34. SecantA line that intersects a circle at two points

References

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3. … ???, … ??? < > ??? < > 2008?.

4. http://www.mathsisfun.com