Geometry

Geometry is an interesting area of Math that requires a proper understanding of the basics. The fundamentals of Triangles, rectangles and rectilinear figures need to be properly taught in order to understand higher concepts.

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Lines Angles and Triangles

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.

Straightlines and Family of Straightlines

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.

Cartesion System of Rectangular Coordinates

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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Coordinate Systems

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.

Angles at a Point

Geometry deals with the study of shape, size, position and other properties of objects in a plane or in space.

Simple Construction

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.

Parallel Lines

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.

Triangles

A triangle is a geometrical figure formed by three lines, which intersect each other and which are not all concurrent.

Isosceles Triangles

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".

Congruent Triangles

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.

Midpoint Theorem

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.

Inequalities (Triangles)

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.

Rectilinear Figures

A plane figure bounded by straight lines is called a rectilinear figure.

Area Theorems

Area of a region, bounded by a geometrical figure measures the portion of the plane occupied by the region.

Parallelograms

We are familiar with plane figures bounded by straight line segments as sides. They are known as Polygons.

Similarity

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.

Loci and Concurrency Theorems

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).

Concurrent Lines in a Triangle

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).

Pythagoras

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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