Right Angles Geometry

Introduction to right angles geometry:

In mathematics, the right angles are right angles triangle in the topic geometry. In geometry, when one of the angles is right angles in any of the three angles then it is called as right angles triangles. In geometry, the right angles have three sides and the two angles with one right angle. In geometry, the right angles have some special cases. The right angles with the special types and the formulas are given below.

Right Angles Geometry – Types:

In geometry, the right angles are classified into two different types. In geometry, the right angles are classified based on only by angles. The classifications of the right angles are given below.

The diagram for the right angles are given as,



Where,

x - Adjacent

y - Opposite

z - Hypotenuse. And

X & Y are angles.

In the right angles, the Pythagoras theorem is the theorem used to give about the right angles.

This theorem is explained by their sides and also the angles in the right angles.

The Pythagoras theorem for sides is given as,

x2 + y2 = z2

The Pythagoras theorem for angles is given as,

Sin A = x / z

Cos A = y / z

Tan A = x / y

The classification of the right angles is given below, Having problem with hard math problems for 12th graders keep reading my upcoming posts, i will try to help you.

Isosceles right angles

Scalene right angles

Isosceles Right Angles:

The structure for the isosceles right angles is given as,



In geometry, isosceles right angles are the right angles where this has two angles be 45 degree and one of an angles is 90 degree in geometry. The ratio of the sides for these types of is (1, 1, v (2)).

Scalene right angles:

The structure for the isosceles right angles is given as,



In geometry, scalene right angles are the right angles where this has one angles be 60 degree and another one is 30 and one 90 degree angles in geometry. The ratio of the sides for these types of is (1, v (3), 2).

Simple Introduction

Introduction to simple introduction of values

The simple introduction position value table by affecting a numeral single place from right to left, its place- value becomes ten times. In the similar method by poignant a digit singles position from left to right, its position-worth becomes one-tenth. So remote we have been discussion concerning seats from right to left beginning from ones place in the simple introduction. Extension of this table from left to right beyond ones place gives rise to the idea of decimals simple. I like to share this Simple Event with you all through my article.

Simple Introduction-basics

The simple introduction concept of decimals is also connected with the fractions denominators are 10, 100 or 1000 etc.

The place value represents the value of the numbers in the actual representations.

The face value represents the starting value of the numbers.

For example:

If we have 45 then the place value for 5 is 5 ones and the face value of 45 is 4

In this situation the educator may help kids to appreciate that a decimal can be spoken as a fraction and vice-versa.

The simple way to represent that "A decimal may surround a entire number module and a decimal module".

In a decimal, if whole number fraction is not specified, we designate it by inscription a zero on the left of the decimal position

For example, 803/100= 0.803;

The thought of simple percent is in use as an integral element of this section and require to be agreed a bit more significance as its utilize in our everyday life is extremely ordinary. Please express your views of this topic how to divide decimals with decimals by commenting on blog.

Basically simple introduction, it is not a original theme but its connection with the learn of fractions and decimals has a particular import. To complicated the notion of percent, we speak that percent earnings for each hundred or and indicate it by the symbol ‘ %’.

Simple Introduction-examples

Permit us think the place value of 4 in the numbers given below:

4365 The place value of 4 is 4000.

3465 The place value of 4 is 400.

3645 The place value of 4 is 40.

and 3674 The place value of 4 is 4.

Slope As A Derivative

In this page we are goimng to discuss about slope as a derivative concept . The trigonometrically tangent of the angle that a line makes with the positive direction of x – axis in anticlockwise direction is called the slope of the line.

The slope of a line in general is denoted by m. Thus m = tan Ø, where Ø is the angle which the line makes with the positive x-axis in anticlockwise direction.

The application of derivatives to plane geometry is related to slopes. Let us consider a curve whose equation is y = f(x).

On this curve we take a point P(x1, y1). Let the tangent at this point not be parallel to the co-ordinate axes, we can get the equation of tangent line at P.

The derivative of the function f(x) is the slope of the tangent.

dy /dx at P(x1,y1) is tan α = slope of the tangent at P, where α is the angle which the tangent at P(x1,y1) makes with the positive direction of x-axis.

The equation of straight line with slope m and passing through the point (x1,y1)is given as

y-y1 = m(x –x1).

For the curve y = f(x) we have  dy /dx =f’(x)

For the tangent line, slope = m = f’(x1)= dy / dx at (x1,y1).

The equation line of the tangent line will be of the form

y-y1 = f’(x1)(x –x1).

If the tangent at P is parallel to the x axis then α =0, `rArr`  tan α =0, `rArr`  slope =dy /dx =0.

If the tangent at P is perpendicular to the x –axis, or parallel to the y-axis then,

α = `\pi/{2}`, `rArr`  tan α =1,  `rArr` slope =dy / dx =1.

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Slope of the tangent  at P(x1,y1) on the curve y = f(x)  is the value of dy/dx at (x1,y1)
Slope of the Normal at P  is perpendicular  to the tangent  and is negative 1/ slope of the tangent.

Slope as a Derivative Examples

Below are the examples on slope as a derivative -

Example 1:
Find the slopes of tangent and normal to the curve y = x4 - 3x2 at (1, -2)
Solution:
Step 1 : Let us write the function y = x4 - 3x2
Step 2 : Then slope is dy/dx = 4x3 - 6x
Step 3 : Slope at (1,-2 ) is 4(1)3 - 6(1) = 4 - 6 = -2  Hence slope of the tangent is m= -2

Step 4 : Slope of the normal = -1/ slope of the tangent = - (-1/2) = 1/2. Hence  slope of the normal = 1/2
Let us find the slope and normal to a function which involves the time element also.

Example 2 :
Find the slope and normal to the curve x = ct and  y = c/t at t
Solution:-
Step 1 : Let us write the equation x = ct

Step 2 : Then the slope  of the function is dx/dt = c
Step 3 :  Let us find dy/dt which is equal to -c/t2
Step 4 :  We have to find dy/dx. Hence here we use chain rule . That is dy/dx = dy/dt ÷ dx/dt = -c/t2 ÷ c = -c/t2 * 1/c
= -1 /t2
Step 5 : Hence slope of the tangent to the curve x =ct , y = c/t is -1/t2
Step 6 : Slope of the normal  at 't'is the reciprocal of the tangent with sign change =  t2
Hence  slope of the normal at 't' is t2
Answer ; Slope of the tangent  at 't' is  -1/t2 and slope pf the normal at 't' = t2

Integrals is the opposite of differential equations.  Hence if slope of an equation is given , then we can find the equation using integrals.
Example 3: Find the equation of the curve whose slope at the point (x,y) is 3x2 + 2, if the curve is passing through the point (1,-1)

Solution:-
Step 1 : We have the slope dy /dx = 3x2 + 2
Step 2 : That gives us dy = (3x2 + 2) dx
Step 3 : Let us integrate both sides ∫dy  = ∫ (3x2+ 2) dx
y  =   3 x3 + 2x + c  = > x3 + 2x + c
3
Step 4 : This passes through (1,-1) Hence we get -1 = (1)3 + 2(1) + c
-1  =  1 +2+c
-1  =   3 + c
That is           c = -4
Step 5 : Let us now write the equation as y = x3 + 2x - 4

Answer :- Equation of the curve whose slope at a point (x,y) is 3x2 + 2 and the point is (1,-1) is y = x3 + 2x - 4
We have used  differential equation  to  find slope of the tangent and slope of the normal to the curve and  we have used integrals  to find the equation of the curve when slope is  given.

Basic Information on Algebra

Introduction for basic information on algebra:

Algebra is the division of mathematics connecting to the study of the regulations of operations and associations, and the organization and ideas occurrence from them, jointly with terms, polynomials, equations and algebraic structures. Collectively with geometry, examination, topology, combinatory, and number theory, algebra is one of the mainly significant divisions of pure mathematics.

Some Basic Information of Algebra:

The following information is very useful for basic algebra.

1. Make a revision of the basic formula with deference to the signs:

(plus +)    +  (plus +)    =   (plus+ )
(minus -)  +  (minus- )  =   (minus -)
(plus +)    *   (plus +)    =   (plus+ )
(minus -)  *   (minus- )  =   (plus +)
(minus -)  *   (plus + )   =   (minus -)
(plus +)    *   (minus- )  =   (minus -)


2. Practicing with the basic formulas:

(x + y) 2 =x2 + y2 + 2xy.
(x - y) 2 =x2 + y2 – 2xy.
x 2 – y 2 = (x + y) (x - y).


3. Attempt the quadratic equation with regard to x and y and solve correspondingly

Example Problems for Basic Information on Algebra:

Example 1 for basic information on algebra:

1) (s + 4) (s – 5)

Solution:

= (s + 4) (s – 5)

= s2 -5s + 4s– 20

= s2 - 1s- 20

Example 2 for basic information on algebra:

2) Solve this algebra expression: 5(a -3) + 4b - 5(a -b -3) + 12

Solution:-

Using Multiply Factors

= 5a - 15 + 4b -5a + 5b + 15 + 12

Groups like terms

= 9b + 12

Example 3 for basic information on algebra:

3) Using algebra equation solve: 3(-5x - 2) - (x - 2) = -2(8x + 4) + 4

Solution:-

Multiplying factors:

-15x – 6 - x + 2 = -16 x - 8 + 4

Group like terms:

-16x – 4 = -16x – 4

Add 16x + 4  to both sides and using the equations as fallows

0 = 0

Example 4 for basic information on algebra:

4) Calculate the value of the following algebra problem: `103^2`

Solution:

Given 1032

= (100+3)2

= 1002+ 2(100) (3) + 32

= 10000 + 600 + 9

= 10609.

Solving Basic Statistics Problems

Introduction to solving basic statistics problems:

Everyday we come across a wide variety of information’s in the form of facts, numerical figures, tables, graphs, etc. These figures are the numerical data collected with a definite purpose, are called data. Data is a word in a plural form of the Latin word datum. Every part of our lives utilizes data in one form or the other. This extraction of significant information is studied in a branch of mathematics called Statistics.

Solving Basic Statistics Problems-statistics Definition

Basically the word ‘statistics’ appears to have been taken from the Latin word ‘status’ meaning ‘a state (related to a political)’. In its origin, statistics is simply the collection of data on different aspects of the life of people, useful to the State. Basic Statistics deals with collection, organization, analysis and Interpretation of data. The word ‘statistics’ has held different meanings in different contexts, so that the calculation process is based on the meaning and data’s.Please express your views of this topic Definite Integrals by commenting on blog.

Solving Basic Statistics Problems
Example for basic stastistics problems:

Consider the marks obtained by 10 pupils in a mathematics test as given below:

55 36 95 73 60 42 25 78 75 62

The data in this form is called raw data.

By looking in this way, can you find the highest and the lowest marks?

Did it take you some time to search for the maximum and minimum marks? Wouldn’t it be a lesser amount of time consuming if these scores were arranged in ascending or descending order? So let us arrange the marks in ascending order as

25 36 42 55 60 62 73 75 78 95

Now, we can clearly watch that the lowest marks are 25 and the highest marks are 95.

The difference of the highest and the lowest marks in the basic stastistics data is called the range of the data. So, the range in this case is 95 – 25 = 70.

Arrangement of data in ascending or descending order can be quite time consuming, particularly when the number of observations in an experiment is large.

Define Rational Function

Introduction to define rational function:

Rational numbers is define as an integer that holds the value which is not equal to 0 and the value form of `x/y` , where x, y is two integers and` y != 0` . Get-together of each and every rational integer(numbers) is denoted by y. A rational integer(numbers) form is said to be proper form, when x and y having no similar factor other than 1. All integers(numbers) are said to be rational numbers. Rational function is referred as f(x). Let us discuss about the rational function in breif.

Define Rational Function:
Rational function is defined as a form that is,

`f(x) = ((p(x))/(q(x)))`

A rational function is generally a division of two polynomial functions mentioned as p(x) and q(x). That is, it is a polynomial divided by an another polynomial. Thus the rational function is defined with the definition and the form of the function is shown. Now let we have some example problems for clear understanding.

Example - Define Rational Function:

Example - Define rational function 1:

Determine the domain for the given rational function f(x) = `(x - 5)/(x^2 + 4x - 12)`

Solution:

Step 1: Given rational function is,

f(x) = `(x - 5)/(x^2 + 4x - 12)`

Step 2: Set the denominator of the given function f(x) = `(x - 5)/(x^2 + 4x - 12)` to zero and solve for x

x2 + 4x - 12 = 0

By rewriting the above equation,

x2 + 6x - 2x - 12 = 0

By separating common terms, we get

x(x + 6) - 2(x + 6) = 0

(x - 2)(x + 6) =0

By factoring each term, we get

x - 2 = 0

x = 2

x + 6 = 0

x = -6

Thus, the domain value of the given function is all real numbers except  for x, x = 2 or - 6

Example - Define rational function 2:

Define the domain of the function f(x) = `(7x)/(18x - 9)`

Solution:

Step 1: Given rational function is,

f(x) = `(7x)/(18x - 9)`

Step 2: Set the denominator of the given function f(x) = `(7x)/(18x - 9)` to zero and solve for x

18x - 9 = 0

By adding 9 on both sides, we get

18x =  9

Divide 18 on both sides, therefore

x = `9/18 `

x = `1/2`

x = 0.5

Therefore, the domain of a given rational function is all real numbers except for x = 0.5.

Solving Basic Exponential Function

Introduction:

Exponential function can be represented as f(x) = bx where b is real number and x is a variable. Domain of any function f(x) is a set of all real values on X-axis for which function is defined and exponential function is defined for all real value of x on X-axis so domain of exponential function is all real number. For function f(x) = bx value of f(x) will be always positive if b is positive and always negative if b is negative so in range of function there will no negative number if b is positive and there will be no positive number in range of function f(x) if a is negative. Exponential functions will be strictly increasing or strictly decreasing depending on value of b. If b >1 then function will be strictly increasing and if b < 1 function will be strictly decreasing for example function y=5x is strictly increasing and function y = (1/5) x is strictly decreasing function. Inverse of exponential function is logarithmic function, logarithmic function can be written as y = logxy where x and y both are positive numbers. Exponential function can also be written as y =ex where e is a real number.



More on Exponential Functions:
For exponential function f(x) = bx , b is called base of exponential function. If two equal exponential functions have same base then their power will be also same for example if bx = by then we can write as x = y, to find inverse of exponential function just inverse the position of x and y in function.Looking out for more help on college algebra solver in algebra by visiting listed websites.

Basic properties of exponential functions:

`e^x e^y = e^(x+y)`
`(e^x)^p = e^(px)`
`de^x/dx = e^x`
`de^(ax)/dx = ae^(ax)`
`d^n e^(ax)/dx^n = a^n e^(ax)`
`e^x/e^y = e^(x-y)`

Example Problem:

Example 1: What will be value of x if 2x = 8?

Solution: 2x = 8

2x = 23

Base is same for both side of equation so

x = 3

Example 2: Find inverse of exponential function y = 5x

Solution: Given function is y = 5x

To find inverse of this function substitute y for x and x for y

x = 5y

log5 x = log5 5y

= y log5 5

= y

So, Inverse of given function is y = log5 x

Example 3: Find the exponential function y = 7x and also graph the function.

Solution: A set of ordered pairs for the equation is shown below.

Putting the value x = -0.5 we get

`y = 7^-0.5`

= 0.3779

(x,y) = (-0.5, 0.3779)

Putting the value x = -0.25 we get

`y = 7^-0.25`

= 0.614

(x,y) = (-0.25, 0.614)

Putting the value x = 0 we get

`y = 7^0`

= 1

(x,y) = (0,1)

Putting the value x = 0.25 we get

`y = 7^0.25`

= 1.626

(x,y) = (0.25, 1.626)

Plotting the given points to get the graph of the exponential function,



This graph shows the part of the exponential function between the co-ordinate points. So, it might not look like a curve but if we plot all the points over the real domain, we get the exact curve.

Proportional Relation

Introduction to proportional relation:

If two quantities vary uniformly such that when one increase the other also increase or when one decrease the other one also decrease then the they are said to be in direct proportion. If two quantities are very that is one increase the other one is decrease then they are said to be inverse proportional relation. In this article we shall discuss proportional relation with example problem.I like to share this Dispersion Relation with you all through my article.

Proportional Relation Example Problem

Example:

A man takes 4 hours to travel 350 miles. How many miles the man travels in 6 hours?

Solution:

Step 1:

Consider the problem as If the person takes 4 hour then the he travel 350miles. If he takes 6 hours then how many miles he travels?

Step 2:

The proportional relationship direct:

4=>350

6=>`6/4 ` x350

=>525

Answer:

Therefore the person traveled 525 miles in 6 hours

Example:

A man purchases 25 articles at `$` 6 each. How many articles can be purchased at the same amount of `$` 50 each?

Solution:

Here the proportions relation is inverse proportion

Cost per article                  Number of article

6                                                            25

50                                                            ?

As cost increase number if article decreases.

Here 6: 50 = x: 25

50 × x = 6 × 25

x = 6 × `25/50` =3

Therefore number of article for the same amount     = 3

Example:

It takes 5 workers 9 hours to repair a machine. How long will it take 12 workers to do the same work they work at the same rate?

Step 1:

Consider the problem as If 5 workers then take 9 hours to complete a job. If 12 workers then how many hour need to repair the same machine?

Step 2:

The inverse proportional relationship:

5 workers => 9 hours

12 workers=>`5/12` x 9

=>`45/12`

Therefore 3.75 hours need to repair a machine

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Proportion Relation Practice Problem

Problem:

It takes 6 workers 9 hours to repair a machine. How long will it take 10 workers to do the same work they work at the same rate?

Answer:

Therefore 5.4 hours need to repair a machine

Problem:

A man takes 6 hours to travel 450 miles. How many miles the man travels in 8 hours?

Answer:

The person traveled 600 miles in 8 hours

Solving a Ratio for a Triangle

Introduction  of solving a ratio for a triangle:

This topic about the ratio in the mathematical concept.

The ratio of width to height of typical computer displays. In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient

Example: For every Spoon of sugar, you need 2 spoons of flour (1:2)

Solving a Ratio for a Triangle Explanation:

In this explanation the ratio of two triangle of same type is the relative between their measures and determines how many times the one triangle is greater than or less than other triangle.

Thus ratio of x and y is the fraction of x / y, and is generally written as x: y



triangle

Here will explain the ratio of the two triangles Based on the value of the each side.

The triangle one value of each side here given,

First triangle AB=4;      BC=5;      CA=7;

Second triangle value is AB=8;  BC=10;   CA=14, for solving this triangle ratio problem.

The Ratio of the two triangle is = first triangle sides / second triangle sides

AB1/AB2;     BC1/BC2;         CA1/CA2;

Final answer of the two triangle is 4/8;       5/10;           7/14;

Ratio=1/2; 1/2; 1/2;

So, Total ratio of two triangle is ½ or 1:2.

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Solving a Ratio for Triangle Problems:
Ex 1:  To solve this Problem the ratio for the two triangles the values are given below?

The triangle one value of each side here given

First triangle AB=3;     BC=5;        CA=6;

Second triangle value is AB=9;   BC=15;        CA=18;

Sol :  By solving ratio of triangle rule,

The Ratio of the two triangle is = first triangle sides / second triangle sides

This formula used for solving this problems.

AB1/AB2;      BC1/BC2;        CA1/CA2;

Final answer of the two triangle is 3/9;      5/15;         6/18;

Ratio=1/3; 1/3; 1/3;

So, ratio of two triangles is ½ or 1:3

Ex 2: To solve this problem the ratio for the two triangles the values are given below?

The triangle one value of each side here given

First triangle AB=2; BC=3; CA=4;

Second triangle value is AB=8; BC=12; CA=16;

Sol : By solving ratio of triangle rule,

The Ratio of the two triangle is = first triangle value / second triangle

This formula used for solving this problems.

AB1/AB2;   BC1/BC2;      CA1/CA2;

Final answer of the two triangle is 2/8; 3/12; 4/16;

Ratio=1/4; 1/4; 1/4;

So, ratio of two triangles is ½ or 1:4

Numerical Data Table

Introduction of Numerical data table:

Numerical data Table shows the measurement of numerical scale. Statistical methods are using for analyzing the numerical data for numerical data table. Values in the numerical data table is using for charts, histogram and graphs. Numerical data table is associating with collection, organization and interpretation of numerical data. In numerical data table, data should be number.

Description of Numerical Data Table:

Numerical data table is incorporated to statistics. The numerical data table is described with frequency distribution, percent and average. Frequency distribution is used for bar graph, Percent is used for pie chart and Average is used for normal graph.

Ordering data in Numerical Data table:

Ordering the data in the numerical data table will be done by qualitative numerical data and quantitative numerical data.

Ordering qualitative numerical data

It shows the numerical data with some quality parameters.

ordering quantitative data.

It shows the numerical data in terms on quantity.

Qualitative data can be shown in:

Pie charts: Another name of pie chart is circle graph, which is partitioned into sector by sector.

Bar graph: Bar graph consists the rectangular bars, which are representing the values.

Quantitative data can be shown:

Statistical tables: This table contains only  statistical data.

Rank order: This is arranging the data by its rank.

Frequency distribution: Frequency distribution denotes data of  intervals

Understanding numerical data table:

Graphs and number understand the values in the numerical data table. We have already seen the different graphs and number above.Is this topic the prime numbers from 1 to 100 hard for you? Watch out for my coming posts.

Example Problems - Numerical Data Table:

1.compute the average of the numbers, 25, 15, 28, 30, 27and 20 on numerical data table.

Solution:

Mean = 25+15+28+30+27+20 / 6

= 145 / 6

= 24.6             

2.compute the Median of the numbers, 15, 20, 28, 25 and 27 on numerical data table.
Solution:

Median = 15, 20, 28, 25, 27 = 27

3.compute the mode of the numbers, 22, 15, 22, 21, 15 and 20 on numerical data table.

Solution:

Here 15 & 22 repeated twice.

So, the modes are 15, 22.

Delta Mathematical Expression

Introduction for delta mathematical expression:

Delta (uppercase Δ, lowercase δ) is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter Dalet. In the Ancient Greek language, it represented a voiced dental plosive /d/, while in Modern Greek it represents a voiced dental fricative, hence it is pronounced as "th" in "that" or "this". it is Romanized as either "D" or "Dh"

(Source: Wikipedia)

Delta Mathematical Expression - Definition Delta and Symmetric Delta:

Definition for delta:

Delta, Δ, and procedures the rate of change of option value with respect to changes in the fundamental asset's price. Delta is the initial derivative of the value V of the derivative with respect to the fundamental instrument's price S.

`Delta ` = `del`V / `del`S   

Symmetric difference for delta:

In symmetric distinction of two sets is the set of basics which are in one of the sets, but not in together. This procedure is the set-theoretic kin of the exclusive disjunction (XOR operation) in Boolean logic. The symmetric distinction of the sets A and B is often denoted by

A`Delta` B

Delta mathematical expression - slope and application:

Delta from slope formula:

Slope = (y2 – y1) / (x2 – x1)

= Δy / Δx

Where, Δ changes in difference

Delta mathematical expression – Example problems:

Example 1:

Find the slope between two points (4,1) (7,6)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(4,1) (7,6) – (x1,y1) (x2, y2)

= (6 – 1) / (7 – 4)

= (5) / (3)

Δy / Δx = 5 / 3

Example 2:

Find the slope between two points (5,6) (7,8)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(5,6) (7,8) – (x1,y1) (x2, y2)

= (8 – 6) / (7 – 5)

= (2) / (2)

Δy / Δx = 1

Delta Mathematical Expression - Application of Delta in Quadratic Equation:

In the polynomials of quadratic equations, Δ is referred as the discriminant value.

Δ = b2 – 4ac

Where Δ is the discriminant value.

Delta mathematical expression - Example problems:

Example 1:

Find the discriminant value given b = 8, a = 2, c = 6

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(6)

= 64 – 48

`Delta` = 16

Example 2:

Find the discriminant value given b = 8, a = 2, c = 4

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(4)

= 64 – 32

`Delta`= 32.

Divide an Angle into Two Equal Angles

Introduction to divide an angle into two equal angles:

Let us assume an angle formed by two rays. The vertex of the angle is the initial points for both the rays. Suppose you need to divide this angle into two equal angles, what is to be done.

One method is to take the help of a protractor to measure the angle. Then you numerically divide that by two, mark a point again using the protractor and join that to the vertex of the angle.

However the better method to divide an angle into two equal angles is by using a compass.

Divide an Angle into Two Equal Angles – Compass Method
The following diagram illustrates the method to divide an angle into two equal angles.



Let OX and OY be the rays making an angle XOY at vertex O.

Set the compass to a convenient radius and strike two arcs from O, one on the ray OX and another on the ray OY. Let the points of intersection of these arcs be A and B respectively. (Fig i)

Now set the compass again for a convenient radius, draw an arc from point A. Without changing the compass setting, draw a similar arc from point B.  Let these arcs intersect at P. Draw a ray connecting the vertex O and the point P. The ray OP divides angle XOY into two  equal angles.  (Fig ii)

The ray OP is called the internal angle bisector of the angle XOY.

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Divide an Angle into Two Equal Angles – Proof for the Method

Join AP and BP (Fig iii)

Considering the triangles  OAO and OBP,

OA = OB  (By construction)

AP = BP  (By construction)

Side OP is common.

Hence the triangles  OAO and OBP are congruent as per SSS postulate.

Therefore , angle AOP = angle BOP (corresponding parts of congruent triangles)

Thus it is proved that the angle XOY is divided into two  equal angles.

Binomial Distribution Normal

Introduction to binomial distribution:
In a specified series of n independent experiments, the discrete probability distribution for the number of successes is called as normal binomial distribution. Bernoulli research is also an experiment of success/failure in normal binomial distribution.Bernoulli experiment is also known as  Bernoulli trial. In normal binomial distribution when n=1, it  is called as Bernoulli distribution.

Mean and Median

Consider the random variable K having the parameters p and n follows the normal binomial distribution. We can write it as K ~ B(n, p).

Probability mass function gives the probability of receiving successes(k) in n trials.

f(k;n,p)=Pr(K=k)=(n k) p^k (1-p)^n-k. for k = 0, 1, 2, ..., n,

where f(k;n,p)=f(n-k;n,1-p)

Mean and variance: If X ~ B(n, p) X  is a binomially distributed variable, then we can say that the expected value of X is E[X]=np. and the variance is Var[X]=np(1-p)

Mode and Median

Normally the mode and the median of a normal binomial B(n,p) distribution is said to be identical to (n + 1)p. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes namely the (n + 1)p and (n + 1)p - 1. However the small special results have been established in the given ratio:

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Mode, median and mean gets coincide for np is the given integer.
Median m should lie in the known interval ?np? = m = ?np?.
For any median m the value cannot recline too far away from the given mean: |m-np| = min {ln2, max {p, 1-p}}.
When p = ½ and n is odd, any m in the given interval ½(n-1) = m = ½(n+1) is said to be a median for the binomial distribution.
If p = ½ and n is the even, then the m = n/2 is an unique median for the given Var(X)=E(X^2)-(E(X))^2. For using the formula need the expected value of X 2:

Geometry Solving Ratios

Introduction to geometry solving ratios

Usually we come across with the ratio of two numbers, if two numbers are x and y then their ratio are given by x : y. Same thing can be extended to geometry of same kind like, between the lengths, areas, perimeters, surface areas, volume etc. These ratios in geometry will help us to know the enlargement or the shrinking of the sizes or shapes of a geometrical figure.

Now let us see few problems of this kind in the following examples.

Example Problems on Geometry Solving Ratios:
Ex 1: The ratio of two supplementary angles are in the ratio 3 : 7. Find the measure of each angle.

Soln: Let the angles be given by 3x : 7x.

Since their sum is supplementary, then 3x + 7x = 180.

Therefore, 10x = 180

`=>` x = 18.

Therefore the angles are 3(18) and 7(18). Hence, they are 540 and 1260.

Ex 2: If the lengths of the sides of a triangle are in the ratio 3 : 4 : 5. Find the area of the triangle if its perimeter is 144cm.

Soln: Given: Perimeter = 144 cm.

The ratios are 3x + 4x + 5x = 144

`=>` 12x = 144

x = 12.

Therefore, the sides of the triangle are given by 3(12), 4(12), 5(12). That is, 36cm, 48cm, 60cm.

To find the area, let us use the Heron’s formula.

Area = `sqrt[s(s-a)(s-b)(s-c)]` . The values of a, b, c can be taken as 36, 48, 60 respectively.

Here S = `[a+b+c]/2`

=`[36 + 48 + 60]/2`

= `144/2` = 72.

Therefore, the area is given by:

A = `sqrt(72(72-36)(72-48)(72-60))`

= 864 cm2.


Between, if you have problem on these topics Nets for Solid Figures, please browse expert math related websites for more help on Types of Sequences.

More Example Problem on Geometry Solving Ratios:

Ex 3: The total surface area of solid cylinder is 616 cm2. if the ratio between its curved surface area and its total surface area is 1 : 2, find volume of the cylinder. [Take `pi = 22/7` ]

Soln: Given: Total surface area = 2`pi` r(r + h)

= 616 cm2 -----------------(1)

Also, the ratios of curved surface area : total surface area = 1 : 2.

`=>` 2`pi` rh : 2`pi` r(r+h) = 1 : 2

`h/(r+h) = 1/2`

2h = h + r

`=>` h = r.

Therefore from (1), we get, 2`pi` r(r+r) = 616

4 `pi` r2 = 616

r2 = 49

r = h = 7.

Therefore, the volume of the cylinder = `pir^2h` = `22/7 xx 7^2 xx 7` = 1078 cm3.                                                

Perimeter of a Right Angle Triangle

Introduction about right angle triangle:

A right triangle or right-angled triangle is a triangle in which one angle is a right angle (that is, a 90 degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The length of boundary is called perimeter triangle. We can calculate the perimeter of right angle triangle by adding the all three sides. In this article we shall discus how to calculate the perimeter of right triangle.

Formula to Find the Perimeter of the Right Triangle:

Perimeter of triangle (P) = a + b + c units

a, b and c are the names of three sides.

Example Problems to Find the Perimeter of the Right Angle Triangle:

1.      Find the perimeter of right angle triangle whose sides are 3 cm, 4 cm and 5 cm.

Solution:

Step 1:  Given:

Side a = 3 cm

Side b = 4 cm

Side c = 5 cm

Step 2:       Perimeter of right angle triangle (P) = a + b + c units

= 3+ 4 + 5

Step 3:        Perimeter of right angle triangle (P) = 12 cm

2.      Find the perimeter of right angle triangle whose sides are 5 cm, 12 cm and 13 cm.

Solution:

Step 1:  Given:

Side a = 5 cm

Side b = 12 cm

Side c = 13 cm

Step 2:     Perimeter of right angle triangle (P) = a + b + c units

= 5 + 12 + 13

Step 3:    Perimeter of right angle triangle (P) = 30 cm

3.      Find the perimeter of right angle triangle whose sides are 9 cm, 12 cm and 15 cm.

Solution:

Step 1:  Given:

Side a = 9 cm

Side b = 12 cm

Side c = 15 cm

Step 2:    Perimeter of right angle triangle (P) = a + b + c units

= 9 + 12 + 15

Step 3:  Perimeter of right angle triangle (P) = 36 cm

I am planning to write more post on What is the Area of a Triangle, Exterior Angle Theorem. Keep checking my blog.

Practice problems to find the perimeter of the right angle triangle:

4.      Find the perimeter of right angle triangle whose sides are 6 cm, 8 cm and 10 cm.

Answer: perimeter of right angle triangle (P) = 24 cm

5.      Find the perimeter of right angle triangle whose sides are 4 cm, 2 cm and 4.47 cm.

Answer: perimeter of right angle triangle (P) = 10.47 cm

6.      Find the perimeter of right angle triangle whose sides are 5 cm, 10 cm and 11.18 cm.

Answer: perimeter of right angle triangle (P) = 26.18 cm

Odd Number Series

Introduction to odd number series:

Odd Number definition: A number ‘x’ which is not divided exactly by 2 then ‘x’ is odd number. For example 3 is odd number which is not divided exactly by 2. If it is divided by 2 then ‘x’ is even number. In another way we can find the odd number easily. Whether the last digit of number ends with 1, 3, 5, 7 and 9 then we can say that as odd number.

For example:  87 are odd or even?

The number end with 7 so according to our definition 87 is odd.

We write the odd number series from 1 to 100 as

1, 3, 5, 7 ... up to 99.
1 + 3 + 5 + 7 + … + 99
To find the nth term of number series formula is

tn = a + (n – 1)d

To find the sum of an number series formula is

Sn = n/2 [ 2a + (n – 1)d ]

Or

Sn = n/2 [ a + l ]

Here ‘n’ is number of term

‘a’ is first term of series

‘d’ is difference between two consecutive terms

‘l’ is Last term of the series

Example: 1 + 3 + 5 + … find 13th term of this series?

Solution:

a = 1;

d = 2nd term – first term = 3 – 1 = 2

n = 13

tn = a + (n – 1)d

= 1 + (13 – 1)2

= 1 + (12)2

= 1 + 24

= 25

t13 = 25

Example Problem on Odd Number Series

Find Sum of first three odd numbers?
Solution:

Let us take first three odd number is 1, 3, and 5.

Sum of three odd number = 1 + 3 + 5

= 9

List out odd numbers greater than 2 and smaller than 20.
Solution:

Which is not divisible by 2 then the number is called odd number.

Odd numbers = 3, 5, 7, 9, 11, 13, 15, 17 and 19.

8 odd numbers available in between the greater than 2 and smaller than 20.

Prove that sum of an even number and odd number as odd number?
Proof:

Let us take ‘2x’ is Even number and ‘2y + 1’ Odd number.

Sum of these even and odd numbers:

2x + (2y +1) = 2(x + y) + 1

Let N = x + y and we can write the sum as

2x + (2y +1) = 2N +1

Let,

x =1 then 2x = 2; y = 2 then 2y + 1 = 5;

2x + 2y + 1 = 2(1 + 2) +1

= 6 + 1

= 7 this is odd number.

Therefore “sum of an even and odd number is odd number”

Sum of three consecutive odd numbers are 21 then find their odd numbers?
Solution:

Let us take 2x + 1 is one odd number

Then consecutive three odd numbers = 2x + 1, 2x + 3, 2x + 5

Sum of these odd numbers = 21

2x + 1+ 2x + 3+ 2x + 5 = 21

6x + 9 = 21

6x = 12

x = 12 / 6

x = 2

Then    2x + 1 = 2 (2) + 1 = 5

2x + 3 = 2 (2) + 3 = 7

2x + 5 = 2 (2) + 5 = 9

The odd numbers are 5, 7, and 9.

3, 7, 11, … find the sum upto 15th term?
Solution:

a = 3; d = 7 – 3 = 4; n = 15

Formula is,

Sn = n/2 [ 2a + (n – 1)d ]

= 15/2 [2*3 + (15 – 1)*4]

= 15/2 [6 + 56]

= 15/2 [62]

S15 = 465

Between, if you have problem on these topics Even Number, please browse expert math related websites for more help on Convert Fraction to Percent.

Exercise on Odd Number Series

Practice problem on odd number series

The sum of two odd consecutive numbers is 16. Find their odd numbers?
Sum of two odd numbers is _______?
The sum of four odd numbers is 40. The three odd numbers is 9, 11 and 13 find the 4th odd number?
1 + 7 + 13 +…. Find the 20th term?
1 + 5 + 9 +….  Find sum up to 18th term?
Answer Key of odd number series

7, 9
Odd
7
115
630