Length of a Line Segment Formula

Length of a line segment formula :   
                                             

Length of line segment is the distance between two coordinate points in a co ordinate plane. It is showed by the units of length .

Suppose there are two points (x₁,y₁) and (x₂,y₂) is

                                D² = (y₂-y₁)² + (x₂-x₁)²

                                This formula is simply a use of Pythagoras' Theorem. 

If the line segments is exactly vertical or horizontal, the formula above will still work fine, but there is an easier way. For a horizontal line, its length is the difference between the x-coordinates. For a vertical line its length is the difference between the y-coordinates. In the figure above make a vertical and horizontal line and verify this for yourself.     

Example on Length of a Line Segment Formula :

problem: Find the distance between two points (length of line segment between two points) in coordinate plane (3,5) , (0,1)

                                       Given two points are (x₁,y₁)= (3,5)

                                                          and (x₂,y₂) = (0,1)

                     There fore

                                                            D² = (y₂-y₁)² + (x₂-x₁)²

                                                            D² =  (1-5)² + (0-3)²

                                                            D² =   (-4)²  + (-3)²

                                                            D² =  16 + 9

                                                            D² = 25

                                                        =>D = √25 = 5

Therefore the length of line segment between the given points is 5 units

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Example on Length of a Line Segment Formula :

Problem: The length of line segment is 13 between the points (1,0) and (a,5) . Find a?

                              Given the length of line segment is 13

                               The points(x₁,y₁)= (1,0)

                               and (x₂,y₂) is (a,5)

                      There fore

                               D² = (y₂-y₁)² + (x₂-x₁)²

                              13²=(a-1)² + (5-0)²

                         => 169= (a-1)² + 5₂

                         => (a-1)² + 25= 169

                         => (a-1)² = 169-25 = 144

                         => (a-1)² = 144

                         => (a-1)²= √144

                         => (a-1)=12

                         => a= 12+1 =13

     The missing value a is 13

Conclusion on Length of a Line Segment Formula :

                                            Finally in this way we can calculate the length of line segment by the formula

  D² = (y₂-y₁)² + (x₂-x₁)²  where

D  is Distance(Length of line segment)

(x₁,y₁) is one point in coordinate plane

(x₂,y₂) is other point in coordinate plane

The length of line segment is denoted by units specified for the points in coordinate plane.

 We can also find the missing points the coordinate axis if the length of line segment is given and one of the points is given. This part of example is shown in example 2 .

Geometry Perpendicular Lines

Introduction:

                   Perpendicular plane or line is a  plane or line  segments interconnect to form a 90 degree (right angle) angle. In geometry, two lines or planes  are measured perpendicular to each one other while they form congruent adjacent angles (a T-shape).  In geometry, we will find out slope of the lines to find the lines are perpendicular or not perpendicular . When the two equations are perpendicular, the products of the slope of  two lines  will be  -1.

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Geometry Perpendicular Lines:

Let us see about geometry perpendicular lines,   

                       In geometry, the line AB is the perpendicular to CD during the point B. By definition, a line is much long, and strictly speaking AB and CD in this example symbolize line segments of two much long lines.Hence the line AB  not having intersect line CD to be measured perpendicular line, because as the line segments are broad away to infinity, they would at rest form similar adjacent angles.

 

Algebra:

In algebra, equation known as y = mx + b, the perpendiculars which having a slope is called as  (-1/m), the opposed reciprocal of the creative slope in geometry. To find the perpendicular of a given line to pass during a point which is y = (-1/m) x + b.

Calculus:

Find the derivative of the function. This will be the slope (m) of every curve at a particular point (x, y) in geometry. Then solving the equation y = (-1/m) x + b

Examples:

1) Find  if y = 3x+7 and x+3y = -5 are perpendicular or not.

Solution:

Equation 1) y = 3x+7 2) x+3y = -5

Slope Intercept Form y = 3x+7 y = (-1/3)x-5

Slope 3 (-1/3)

Since the product of the slopes is [3*(-1/3)]= -1, the two lines are perpendicular.

2) Find  if y = 3x+4 and 4x+2y = 4 are perpendicular or not.

Solution:

Equation 1) y = 3x+4 2) 8x+4y =8

Slope Intercept Form y = 3x+4 y = -2x+2

Slope 3 -2

Since the product of the slopes [3 * (-2)] does not equal -1 , the two lines are not perpendicular.

Introduction to decimal place values

Introduction to decimal place values:

         The decimal place value of number is determined by its places. The numbers are placed in the exacting number of a digit within a numerical numbers. The place values point to the decimal place of a numerical system base value. The values are showing that numerical value in the standard forms in the recognition. The values are determined through place of the numerals availability.

Explanation of Decimal Place Values :

Decimal place values is given below that

Words -- Numbers-

       Tenth place denote 0.1

       Hundredth place denote 0.01

       Thousandth place denote 0.001

       Ten thousandth places denote 0.0001

        One hundred thousandth place denote0.00001

        One million place denote 0.0000001

        Ten million place denote 0.0000001

         Hundredth million place denote 0.00000001

Representation of decimal place value chart decimals

       

Number showed, such as .25, contain two numbers. Every digit has a different place value.

The after decimal place number indicate the tenths place. So, indicated as 1 tens.

The next tenths place number is the hundredths place which is 5 in this example. So, there is 2 set of 10, in addition 5 hundredths in the number 0.25.

The number .44 might be representing by this table:

The number 44 is the combination of tenths and hundredths place value

This number contains 4 tenths and 4 hundredths

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Math Examples for Decimal Number Place Value:

Decimal numbers, such as O.1234, contain four numbers after the decimal point.  Each number is a different place value. 

The initial number after the decimal point number is called as tenths place value. There is one tenths in the number O.1234.

The second number indicates through how many hundredths there in the number.  The number O.1234 has two hundredths.

The third number is the thousandths position.

The fourth number is recognizing that the ten thousandths position which is four in the example.

The one tenths, two hundredths, three thousandths, and four ten thousandths position in the number of 0.1234.so, this number is combination of  1 tenths and 2 hundredths ,3 thousandths and 4 ten thousandths.

Introduction to class of functions

Introduction to class of functions:

·         In mathematics, the function means that assigns element from one set into another set element. The sets may be sets of numbers, sets of points, and sets of objects.

·         All the function has to be to do is to assign some element from the second set to each element in the first set.

·         Function can be classified two groups. Algebraic function and transcendental functions.

Algebraic Class Function:

                           A function class which involves operations only of additon, subtraction, multiplication and exponents on an  independent variable x is called an algebraic function.

Three particular class of functions are:

Polynomials function:

                A function of the form a₀xⁿ+a₁x^n-1+a₂x^n-2+........an where n is a positive integer and a₀,a₁,a₂...an are real constants is called a Polynomials function of a x.

Rational functions:

                  A function class of the form f(x)/g(x) where f(x) and g(x) are polynomial in x is called rational functions.

Irrational functions:

                The functions involving radicals are called irrational functions.

 

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Transcendental Class Function:

                             The following different class of  transcendental function are given as,

·         Trigonometric functions, such as cos x, tan 3x

·         Inverse trigonometric functions ,such as sin^-1x, sec^-1x

·         Logarithmic functions: The correspondence x-->log x is a logarithmic functions.

Identity function:

·         The identity function maps each member of the domain onto itself. That is, if I is the identity function ,then for any x in the domain, I(x)=x

·         If we let y stand for I(x), then y=x. It is easy to see that if the domain of I is the set of real number ,then the graph of the identity function is the line y=x. The identity function receives its name because it is the identity for the operation of composition.

Even and odd function:

·         If f (x) is a function of x such that f(-x)=f(x) for the every x of of its domain ,then it is called an even function of x

·         If f (x) = cos (x) ,then f(-x)= cos (-x)=cos x so that f (x)=f (-x).Thus f(x)=cos x is a even function.

·         If f (x) is a function of a x such that f (-x)=-f (x) for every x of its domain, then it is called an odd function of x.

Introduction to arithmetic maths

Arithmetic or arithmetic is the oldest and mainly basic branch of mathematics, used by approximately everyone, for tasks range from simple day-to-day including to superior science and business computation. It engages the learn of quantity, particularly as the outcome of unite numbers. In general usage, it refers to the easier properties when using the conventional operations of adding, subtracting, multipling and dividing with lesser values of numbers.

Basic Operations for Arithmetic Maths:

The arithmetic maths operations are as follows,

         1. Addition.

         2. Subtraction.

         3. Multiplication.

         4. Division.

Arithmetic Math- Addition:

Calculation of two or more than two numbers is known Arithmetic. The indication for addition is +. We can execute the addition operation for the all kind of values. Addition problems augment the values.

Arithmetic math Addition- Example:

          6 + 5 = 11

         This is easy addition problem.

         Similarly we do the addition operation also more than two digits.

         25 + 33 = 58

Arithmetic math- Subtraction:

Subtraction is the one of the essential arithmetic operation. Reduction of two or more than two numbers is known as Subtraction. The sign of subtraction is - (minus). Subtraction problems reduce the values.

         The subtraction is finished by the following way,

          z = x – y

          Where,

          z is differentiation.

          x is known as minuend value.

          y is known as subtrahend value.

Arithmetic maths subtraction- Example:

         8-5 = 3.

Arithmetic Maths- Multiplication:

Multiplication is the calculation of two numbers. The indication of multiplication is *. Multiplication also execute by recurrence of calculation. The conditions often used for multiplication are “times”.

Arithmetic maths multiplication- Example:

       4 times 8 means 4 multiply with 8.

       4 * 8 = 32

      The other technique of multiplication recurring addition,

       4 times 8 = 32

       8 + 8 + 8 + 8 = 32

Arithmetic maths- division:

Division is an annul of multiply operation. The indication of division is /. The major reason of division is reducing the value and it is also recurring calculation.

        Appearance of division technique,

        `x/y` =z.

        Where,

         x is called as dividend.

         y is called as divisor.

         z is called as quotient.

Arithmetic math division- Example:

        `21/7` = 3.

Introduction to Arithmetic Series Formula

A sequence is a list of numbers. A series is created by adding terms in the sequence. This lesson assumes that you know something about arithmetic sequences, how to find the common difference and how to find an explicit formula. You may want to review the sequences or finding formulas. There are two ways to denote that you are adding terms in a sequence. One method is by using summation notation and another one method is by using subscript notation similar to how we write explicit forms of sequences.

General Formula for Arithmetic Series:

Arithmetic progression is also called arithmetic sequence, is a sequence that begins with an initial term a, and then each term is found by adding the common difference d.

General Form of arithmetic sequence is,
a, a + d, a + 2d, a + 3d + . . .

The recursive formula is,

an = an-1 + d.

Here, d -> common difference,

n -> number of terms.

The explicit or closed form of an arithmetic sequence is,

an = a1 + (n - 1) d.

Example Using Arithmetic Series Formula:

Ex 1: Find the sum of an = (`(1)/(2)` )n + 1, where nth term is 25.

Sol:  The 25th partial sum of this series is the sum of the first twenty-five terms. The first few terms of the sequence are:

a1 = `(1)/(2)` `xx` 1 + 1 = `(3)/(2)`

a2 = `(1)/(2)` `xx` 2 + 1 = 2

a3 = `(1)/(2)`  `xx` 3 + 1 = `(5)/(2)`

These terms have a common difference d = `(1)/(2)` , so this is indeed an arithmetic sequence. The last term in the partial sum should be,

a35 = a1 + (25 – 1)(d)

=> a35 = `(3)/(2)` + `24 xx (1)/(2)`

=> a35 = `(27)/(2)`

Then, plug the value of n in the formula, the 25th partial sum is:

`(n)/(2)` (a1 + an) =  ` (25)/(2) xx ((3)/(2) + (27)/(2))` = `(25)/(2) xx (30)/(2)` = `(375)/(2)`



Ex 2: Find the nth term of the arithmetic sequence 1, 4, 7…

Sol:    an = a1 +(n – 1) d

=> 1 + (n – 1) (3)

=> 1 + 3n – 3

=> 3n – 2.



Ex 3: The first term of an arithmetic sequence is 5 and the eighth term is 19. Find the 20th term.

Sol:    an = a1 +(n – 1) d

=> a5 = a1 + (8 – 1) d

=> 19 = 5 + 7d

=> 14 = 7d

=> d = 2

Therefore 20th term is,

=> a20 = 5 + (20 – 1) (7)

=> a20 = 5 + (19) (7)

=> a20 = 5 + 133

=> a20 = 138.

Introduction to Prism volume equation

Introduction to Prism volume equation:

                        A prism is a solid and that has two parallel faces which are congruent polygons at both ends. This faces are formed the base of the prism. A prism is named after the shape of its base side. The volume of a prism depends on their sides. A triangle pyramid has a triangle for a base.

There are two types of prism based on their base side that is shown below
1. Rectangular prism
2. Triangular prism
The volume of a prism is given

        Triangular prism equation for volume V = * length * width * height.          
Where
L is the length of the prism.
W is the width of the prism
H is the height of the prism

Rectangular prism:
        Rectangular prism equation for volume V = a * b * h
Where
a – Apothem length
b – Breadth
h – Height

Example Problem for Triangular Prism Volume Equation:

1.Solve the volume of the right prism base length 8 height 12 and width 9.

Solution:

            Given that Length  = 8

                              Width   =  12

                               Height = 9

Formula for volume of a triangular = `1 / 2`  * length*width*height.

                                                             = `1 / 2`  *  8 * 12 * 9

                                                              =`1 / 2` * 864

                                                              = 432

The solution is = 432

2.Solve the volume of the right prism base length 12 height 20 and width 14.

Solution:

            Given that Length  = 12

                              Width   =  20

                               Height = 14

Formula for volume of a triangular = `1 / 2` * length*width*height.

                                                            = `1 / 2`  * 12 * 20 * 14

                                                            = `1 / 2 `  * 3360

                                                            = 1680

The solution is = 1680

Example Problem for Rectangular Prism Volume Equation:

3.The prism has its apothem length 9 cm , base side 2cm and height 7cm. find the volume

Solution:

Given that  a - 9 cm

                   b – 2 cm

                   h – 7 cm 

Formula for volume of a triangular pyramid = `(1/6)` abh

                                                                              = `1/6`  * 9 * 2 * 7

                                                                             = `1/ 6 ` * 126

                                                                             = 21

The solution is =21

4.The prism has its apothem length 0.65 cm , base side 0.23cm and height 0.82cm. find the volume

Solution:

Given that  a – 0.65 cm

                   b – 0.23 cm

                   h – 0.82 cm 

Formula for volume of a triangular pyramid = `(1/6)`  abh

                                                                        = `1 / 6`  * 0.65 * 0.23 * 0.82

                                                                        = 0.122

The solution is = 0.122