Geometry: Concepts & Basics

Triangles 

The plane figure bounded by the union of three lines, which join three non collinear points, is called a triangle.
There are several special types of triangles with important properties.  But one property that all triangles share is that the sum of the lengths of any two of the sides is greater than the length of the third side or difference of the lengths of any two of the sides is less than the length of the third side, as illustrated below.

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Triangles are similar if they have the same shape, they may be of different sizes. Therefore, the triangle obtained by rotating a triangle is similar to the original triangle. Similarly, the mirror image or the water image of a particular triangle is similar to the original triangle. 
A triangle is defined by particularly six measures (three angles and three sides). But we don't need to know all of these to show that two triangles are similar to each other. 
Triangles are similar if:
  1. AAA (angle angle angle): Two triangles are said to be similar if two pairs of corresponding angles are equal. This also means that the remaining pair of the angles will be congruent as well.
  2. SSS (side side side): Two triangles are also said to be similar if, all the three pairs of corresponding sides are in the same proportion to each other.
  3. SAS (side angle side): Two triangles are said to be similar if, two pairs of sides are in the same proportion and the included angles are equal to each other.
When two triangles are similar, the following properties always hold true:
  • Corresponding angles are equal to each other (same measure)
  • Corresponding sides are in the same proportion to each other.
Pythagoras Theorem : 
The Pythagoras' theorem, states that, “the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides”. The theorem can also be written as an equation relating the lengths of the sides x, y and z, often called the "Pythagorean equation”. Although Pythagoras theorem is applicable in case of a right-angled triangle only, yet it has a lot of direct and indirect applications. The basic formula to calculate hypotenuse is x2 + y2 = z2, where z represents the length of the hypotenuse and x and y the lengths of the triangle's other two sides.
Classification of triangles on Pythagoras theorem: 
  • If x2 + y2 < z2, then the triangle will be an obtuse angled triangle.
  • If x2 + y2 > z2, then the triangle will be an acute angled triangle.
Circles : 
A circle is a set of points in a plane that are all located at the same distance from a fixed point (the center of the circle).
A chord of a circle is a line segment that has its endpoints on the circle.  A chord that passes through the center of the circle is a diameter of the circle.  A radius of a circle is a segment from the center of the circle to a point on the circle.  The words "diameter" and "radius" are also used to refer to the lengths of these segments.
Polygons : 
A polygon is a closed figure formed by three or more than three line segments. These line segments are called the sides of the polygon. Each side of the polygon intersects exactly two other sides at their respective endpoints. The points of intersection of the sides of the polygon are known as vertices of the polygon. The term "polygon" is used to refer to a convex polygon, that is, a polygon in which each interior angle has a measure of less than 180.
Convex polygon: A type of polygon in which none of the interior angles is more than 180, is called a convex polygon.
Concave polygon: A type of polygon in which at least one of the interior angles of the polygon is more than 180, is called concave polygon. 
Regular polygon: A type of polygon which has all its angles and sides equal is called a regular polygon.
A polygon with three sides is called a triangle; with four sides, is called a quadrilateral; with five sides, is called a pentagon; and with six sides, is called a hexagon.
The sum of the all the interior angles of a triangle is always 180.  
 The sum of all the interior angles of any polygon with n sides is equal to (n - 2)180.  For example, the sum for a hexagon is (6 - 2)180 = (4)180 = 720.
Note that, a hexagon can be divided into four triangles, therefore, the sum of the angle measures of a polygon can be found by adding the sum of the angle measures of four triangles. The perimeter of any polygon is the sum of the lengths of its sides. The phrase "area of a triangle" (or area of any other plane figure) is used to mean the area of the region enclosed by that particular figure.
Properties of polygons
  • If number of sides of the polygon is n, then the sum of all the interior angles = (n-2)180.
  • Interior angle of a polygon + corresponding exterior angle of that polygon = 180.
  • The sum of all the exterior angles of a polygon = 360°.
  • The number of the  diagonals in a polygon of n sides is always  = (n(n-1)/2) - 1
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