CLASS-7
CONCEPT OF TRIANGLES

CONCEPT OF TRIANGLES -

The concept of a triangle is fundamental in geometry. A triangle is a polygon with three edges (sides) and three vertices (corners). The most basic properties of triangles stem from these three sides and three angles. Here's a breakdown of the key aspects of triangles:

1. Definition:-

A triangle is a closed, two-dimensional figure made up of three straight sides that are connected at three vertices, forming three interior angles. The sum of the interior angles of any triangle is always 180°.

2. Classification of Triangles:-

Triangles can be classified based on their sides or angles:

   A) By Sides:-

  • Equilateral Triangle:-  All three sides are of equal length, and all angles are 60°.
  • Isosceles Triangle:- Two sides are of equal length, and two angles are equal.
  • Scalene Triangle:-  All three sides have different lengths, and all angles are different.

   B) By Angles:-

  • Acute Triangle:-  All three interior angles are less than 90°.
  • Right Triangle:- One angle is exactly 90° (a right angle).
  • Obtuse Triangle:-  One of the angles is greater than 90°.


3. Properties of Triangles:-

  • Angle Sum Property:- The sum of the interior angles of a triangle is always 180°.
  • Exterior Angle Theorem:- The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
  • Triangle Inequality Theorem:- The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
  • Pythagoras Theorem (for right triangles):- In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.


4. Types of Special Triangles:-

  • Right-Angled Triangle:-  Contains a 90° angle, and the Pythagorean theorem applies.
  • Equilateral Triangle:-  All sides and angles are equal, making it a highly symmetric figure.
  • Isosceles Triangle:- Has two equal sides, leading to two equal angles.
  • Scalene Triangle:- No sides or angles are equal, making it asymmetrical.


5. Formulas Related to Triangles:-

  • Area of a Triangle:-

                                 1

     Basic formula:-    A = ----- × base × height

                                 2

     Heron’s formula (when all sides are known):-  A = √s(s−a)(s−b)(s−c)

                                              (a+b+c)      

     where s is the semi-perimeter, s = ---------, and a,b,c are the 

                                                 2

       sides of the triangle.

     Perimeter of a Triangle:-

     The perimeter is the sum of the lengths of its sides:  P = a+b+c


6. Centers of a Triangle:-

  • Centroid:- The point where the three medians (lines drawn from each vertex to the midpoint of the opposite side) intersect. It is the triangle's center of gravity.
  • Circumcenter:- The point where the perpendicular bisectors of the sides intersect. It is the center of the circumcircle (the circle that passes through all three vertices).
  • Incenter:- The point where the angle bisectors intersect, which is the center of the incircle (the circle that touches all three sides from the inside).
  • Orthocenter:- The point where the three altitudes (the perpendiculars from each vertex to the opposite side) intersect.


7. Applications of Triangles:-

  • Architecture and Engineering:- Triangles are used for structural strength due to their rigidity.
  • Trigonometry:- The study of triangles leads to the development of trigonometric ratios (sine, cosine, tangent) used in various fields like physics and astronomy.

Triangles form the basis of many advanced geometric concepts and are fundamental in understanding shapes, angles, and spatial reasoning.