FUNDAMENTAL CONCEPTS

1. Results on Triangle

1. Sum of the angles of a triangle is 180°.

2. The sum of any two sides of a triangle is greater than the third side.

3. Pythagoras Theorem:

   In a right-angled triangle, (Hypotenuse)² = (Base)² + (Height)².

4. The line joining the mid-point of a side of a triangle to the opposite vertex is called the median.

5. The point where the three medians of a triangle meet, is called centroid. The centroid divided each of the medians in the ratio 2 : 1.

6. In an isosceles triangle, the altitude from the vertex bisects the base.

7. The median of a triangle divides it into two triangles of the same area.

8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle.

9. Results on Quadrilaterals:

   1. The diagonals of a parallelogram bisect each other.

   2. Each diagonal of a parallelogram divides it into triangles of the same area.

   3. The diagonals of a rectangle are equal and bisect each other.

   4. The diagonals of a square are equal and bisect each other at right angles.

   5. The diagonals of a rhombus are unequal and bisect each other at right angles.

   6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

   7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

IMPORTANT FORMULAE

1. 1.   Area of a rectangle = (Length x Breadth)

2.     Length = \([\frac { Area } { Breadth } ]\)and Breadth =\([\frac { Area } { Length } ]\)

3. 2.   Perimeter of a rectangle = 2(Length + Breadth).

4. Area of a square = (side)² =  \( \frac { 1 } { 2 } \)(diagonal)².

5. Area of 4 walls of a room = 2 (Length + Breadth) x Height.

6. 1.   Area of a triangle =  x \( \frac { 1 } { 2 } \)Base x Height.

   2.   Area of a triangle = s(s-a)(s-b)(s-c)
         where a, b, c are the sides of the triangle and s = \( \frac { 1 } { 2 } \)(a + b + c).

7.   3 . Area of an equilateral triangle = \( \frac { \sqrt3} { 4 }\) x (side)^2.

    4. Radius of incircle of an equilateral triangle of side a =\( \frac { a} { 2 \sqrt3 }\)

   5.  Radius of circumcircle of an equilateral triangle of side a =\( \frac { a} { \sqrt3 }\)

    6. Radius of incircle of a triangle of area  and semi-perimeter r =/s

   VIII. 1.Area of parallelogram = (Base x Height).

            2.  Area of a rhombus =\( \frac { 1 } { 2 } \) x (Product of diagonals)

            3 .Area of a trapezium = \( \frac { 1 } { 2 } \)x (sum of parallel sides) x distance between them

          1.   Area of a circle = R², where R is the radius.

          2.   Circumference of a circle = 2R 

8.        3. Length of an arc = \( \frac {2 \pi R} { 360 } \)where  is the central angle.360

9.      Circumference of a semi-circle = R.