### Dictionary Rank of a Word | Permutations & Combinations

PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

### Types of Quadrilaterals, their properties, shapes, and areas

Types of Quadrilaterals, their  properties, shapes, and areas

Any closed figure bounded by four sides is called a quadrilateral. A quadrilateral has four sides four vertices and four angles.

A quadrilateral have 4 sides, 4 vertices and 4 angles.
Sum of all angles of a quadrilateral is 360

$Area\: of\: quadrilateral\: =\frac{1}{2}\times AC(DE+BF)$
Trapezium:-

If one pair of opposite sides of a quadrilateral is parallel then it is called a trapezium.

Properties of Trapezium

One pair of opposite sides are parallel.
Two pair of adjacent angles are supplementary.
$Area\: of\: trapezium=\frac{1}{2}\times (b_{1}+b_{2})\times h$

Parallelogram:-
If both pair of opposite sides of a quadrilateral is parallel then the quadrilateral is a parallelogram.
If one pair of opposite sides of a quadrilateral is equal and parallel then the quadrilateral is a parallelogram.
If both pair of opposite sides of a quadrilateral are equal and diagonals are not equal then the quadrilateral is a parallelogram.
Properties of parallelogram
• Opposite sides are equal and parallel.
• Opposite angles are equal.
• Diagonals bisect each other.
• Diagonals are not equal.
• Area of parallelogram = Base x Height
Rectangle:-
If one angle of a parallelogram is right then it is called a rectangle.
If both pair of opposite sides of a quadrilateral are equal and diagonals are also equal then the quadrilateral is a rectangle.
Properties of Rectangle

• Both pair of opposite sides are equal and parallel.
• Diagonals are equal and bisects each other.
• All angles are equal to  90o
Area of rectangle = Length X Breadth

Rhombus:-
If adjacent sides of a parallelogram are equal then it is called a rhombus.
If all sides of a quadrilateral are equal and diagonals are not  equal then the quadrilateral is a rhombus.
Properties of Rhombus:-

• All sides are equal.
• Opposite sides are equal and parallel.
• Opposite angles are equal.
• Diagonals  bisects each other at right angle.
• Diagonals are not equal.
• Area of Rhombus = Base x Height.
• $Area\: of\: Rhombus\: =\frac{1}{2}\times Product\: of\: diagonals$
• $Area\: of\: Rhombus\: =\frac{1}{2}\times d_{1}\times d_{2}$
Square:-
A parallelogram whose one angle is right and adjacent sides are equal is called a square.

A rectangle whose adjacent sides are equal is called a square.

A rhombus whose one angle is right is called a square.

If all sides of a quadrilateral are equal and diagonals are also equal then the quadrilateral is a square.

### Properties of Square

• All sides are equal.
• All angles are equal to 90o
• Diagonal are equal.
• Opposite side are equal and parallel.
• Opposite angles are equal.
• Perimeter of square is = 4 x side
• Area of Square is = (side x side)
• $Area\: of\: Square\: =\frac{1}{2}\times d_{1}\times d_{2}$
• $Area\: of\: Square\: =\frac{1}{2}\times product\: of\:diagonals$

Kite

If two pair of adjacent sides of a quadrilateral is equal then the quadrilateral is a kite.

### Properties of Kite

• Two pairs of adjacent sides are equal.
• Diagonals are perpendicular to each other.
• One diagonal bisect the other.
• $Area\: of\: kite\: =\frac{1}{2}\times d_{1}\times d_{2}$

 Type Figure Perimeter Area Quadrilateral Sum of four sides $\frac{1}{2}\times AC[DE+BF]$ Trapezium Sum of four sides $\frac{1}{2}\times (b_{1}+b_{2})\times h$ Parallelogram Sum of four sides Base x Height Rectangle 2 (Length + Breadth) Length x Breadth Rhombus 4 x Side $\frac{1}{2}\times d_{1}\times d_{2}$ Square 4 x Side $=side\times side\\\\ \: \:\: \: \: \: =\frac{1}{2}\times d_{1}\times d_{2}$ Kite Sum of 4 sides $\frac{1}{2}\times d_{1}\times d_{2}$

• Quadrilateral formed by joining the mid points of the sides of quadrilateral is called a parallelogram.
• Quadrilateral formed by joining the mid points of the sides of  a Parallelogram is called a Parallelogram.
• Quadrilateral formed by joining the mid points of the sides of Rectangle is called a Rhombus.
• Quadrilateral formed by joining the mid points of the sides of  Rhombus is called a Rectangle.
• Quadrilateral formed by joining the mid points of the sides of Square is called a Square.
• Quadrilateral formed by joining the angle bisectors of opposite vertices of a parallelogram is called a Rectangle.
Polygons:-
Closed figures with sides three or more are called polygons.

Polygon with  three sides is called triangle.

Polygon with four sides is called Quadrilateral.

Polygon with five sides is called Pentagon.

Polygon with six sides is called Hexagon.

Polygon with seven sides is called Heptagon. and so on....

Sum of all interior angles of a polygon is given by the formula = (n-2) X 180.

Measure of each interior angle of a polygon = $\left ( \frac{n-2}{n} \right )\times 180^{o}$

Measure of each exterior angle of a polygon  =  $\frac{360^{o}}{n}$

Sum of all exterior angle of a polygon = 360o

Number of diagonals of a polygon of side n is =  $\frac{n^{2}-3n}{2}$  or  $\frac{n(n-1)}{2}-n$