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Geometry is a subject which starts from a point, goes to a straight line to planes, curved lines, curved surfaces and on to solid objects. It is a science where the answers to basic questions are only approximations but that does not hinder the most abstract concepts. Geometry needs to be taught properly and TutorVista has a wealth of tutors who are masters at this. Learn how to prove theorems, axioms and how to construct polygons, circles under expert tutelage. Get comfortable with the Bolas Para or Hyper. Geometry tutoring at TutorVista will guide you in coming out of the complex labyrinths of geometry with flying colors
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Lines Angles and Triangles |
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Geometry originated when man felt the need to measure his land. Ancient Egyptians were perhaps the first people to study geometry. Later, the Babylonians studied in a systematic way. |
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Straightlines and Family of Straightlines |
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A rational, integral, algebraic equation of the variables x and y is said to be homogeneous equation of nth degree in x and y, when the sum of the indices of x and y in every term is the same and is equal to n. |
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Cartesion System of Rectangular Coordinates |
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Rene' Descartes' (1596-1665), a French philosopher and mathematician, introduced a method by which the position of a point can be corresponded with an ordered pair of real numbers. These pair of real numbers are called the Coordinates. This method is the new idea of combining two branches of mathematics, Algebra and Geometry. The combination of these two branches of mathematics was called Algebraic Geometry, Coordinate Geometry or Analytical Geometry. |
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Angles at a Point |
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Geometry deals with the study of shape, size, position and other properties of objects in a plane or in space. |
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Simple Construction |
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Geometrical figures help us to understand various geometrical concepts. When we prove geometrical propositions by logical reasoning, we draw only a rough figure and we do not need to take accurate measurements but geometrical constructions have to be drawn accurately to the given measurements. They are used by scientists, artists and engineers. These constructions are done using ruler and compass only. |
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Parallel Lines |
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Consider two lines in a plane. They either have one point in common, or they have no common point. When they have a common point, they are called intersecting lines. When they have no common point, they are called parallel lines. The distance between two parallel lines is the same at all points. |
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Triangles |
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A triangle is a geometrical figure formed by three lines, which intersect each other and which are not all concurrent. |
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Isosceles Triangles |
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Triangles are classified on the basis of the lengths of their sides as scalene, isosceles, or equilateral triangles. When any two sides of a triangle are equal it is called as an "isosceles triangle". The unequal side is called its base and the angle opposite the base is called the "vertical angle". |
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Congruent Triangles |
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Every geometric figure has a shape and a size. Two circles of different radii have the same shape but their sizes are different. But if we draw two circles of the same radius both the shape and size will be the same. Such figures with the same shape and size are called congruent figures. We can check whether two figures are congruent or not by the method of superposition. |
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Midpoint Theorem |
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We can prove some more properties of triangles using the properties of parallelograms seen in the previous chapter. We find that the line segment joining the mid points of any two sides of the triangle is parallel to the third side and is equal to half of it. We prove this in the mid point theorem. |
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Inequalities (Triangles) |
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We have studied so far about equalities in a triangle. In an isosceles triangle two of the sides are equal. In an equilateral triangle all the sides are equal. But there exist several situations where we need to compare quantities which are not equal. This gives rise to the concept of inequalities. |
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Rectilinear Figures |
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A plane figure bounded by straight lines is called a rectilinear figure. |
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Area Theorems |
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Area of a region, bounded by a geometrical figure measures the portion of the plane occupied by the region. |
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Parallelograms |
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We are familiar with plane figures bounded by straight line segments as sides. They are known as Polygons. |
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Similarity |
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A perpendicular drawn from the right angle vertex of a right triangle to the hypotenuse divides the triangle into two triangles similar to each other and also to the original triangle. |
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Loci and Concurrency Theorems |
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Sometimes it is necessary in geometry to specify the location of all points satisfying one or more conditions. For example, location of set of points equidistant from two given points, location of set of points equidistant from a given point etc. This is done by considering a set of points that satisfies the given condition(s) and then testing that every point that satisfies the given condition(s) is in the set. Here are a few examples for locating set of points that satisfy the given condition(s). |
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Concurrent Lines in a Triangle |
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Draw a triangle ABC. Draw the perpendicular bisectors of its sides. The perpendicular bisectors of the sides of a triangle are concurrent (pass through the same point). |
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Pythagoras |
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In a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the sides containing the right angle. |
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