Alternate Segment Theorem

Given a circle and two points P and Q on its boundary and a departure to the circle at Q, then angle between the departure and the line PQ is the same as the angle subtended by this chord in segment of the circle on the opposite side of PQ.  Alternate segment theorem states that approach involving a departure and its chord is equal to angle in the alternate segment.Interactive animatronics representatives proof of an alternate segment theorem.

Theorem for alternative segment

         The length of the line segment AB, which joins A (x₁, y₁) and B (x₂, y₂) is given by
d= `|AB|=sqrt((x2-x1)^(2)+(y2-y1)^(2))`  
Proof:
Let A (x₁, y₁) and B (x₂, y₂) be two points in the plane.
Let d = distance between the points A and B.
Draw AL and BM perpendicular to x-axis (parallel to y-axis).
Draw AC perpendicular to BM to cut BM at C.

OL = x₁, OM = x₂ [AC = LM = OM - OL = x₂ - x₁]
MB = y₂, MC = LA = y₁ [CB = MB - MC = y₂ - y₁]
`AB^(2)=AC^(2)+CB^(2)`
`d^(2)=(x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)`
d=`sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1)^(2)))`

Alternate Segment Circle Theorem

           The chord CE divides the circle into 2 segments. Angle CEA and CDE are angles in alternate segments because they are in opposite segments. Circles are easy blocked curves which separate the plane into two regions, an interior and an exterior.The alternating segment theorem states that a position connecting a departure and a chord through the point of contact is equal to the angle in the alternate segment.
         In conditions of the beyond diagram, the alternating segment theorem tells us to facilitate angle CEA and angle CDE are equivalent.



A digression makes an angle of 90 degrees with the radius of a circle,so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180.
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90.
But OAC + x = 90, so ∠OAC + x = ∠OAC + y.
Hence x = y.

Fractions Addition

The whole thing is partitioned in two different pieces. Each piece is called fraction of the whole thing.A fraction represents the divided form of two numbers. It takes the general form of `(a)/(b)` . Here a represents the numerator value and b represents the denominator value. `(17)/(5)` is an example for fraction. Fractions are classified according to their values numerator and denominator which plays the important role in fractions addition table

Types of fraction

1.Proper Fraction

2.Improper Fraction

3.Mixed Fraction

4.Equivalent Fraction


Activities involved in fractions addition:

1. When we add two fractions with same denominator, those fractions are added directly.

For example:

`(2/3) + (7/3) = (2+7)/(3) = 9/3`   (here numerator is divided by denominator)

=3

2. When we add two fractions with different denominator, we have to find least common denominator.

`(2/3) + (6/9) = [(2xx3)+6]/(9) = (6+6)/(9) = (12)/(9)` (here numerator is divided by denominator)

=4/3

3. If mixed fraction is involved in fractions addition, we have to convert the mixed fraction to improper fraction. Then we have to check the denominators. If the denominators are different, we have to find least common denominator. Otherwise no problem.

2 ¾ + 5/4

Take the mixed fraction 2 ¾

Convert it to improper fraction as below

` [(2xx4)+3]/(4) = (8+3)/(4) = (11)/(4)`

Now we can do the fractions addition

`(11)/(4) + (5)/(4)= (11+5)/(4) = (16)/(4) = 4`

About Easy Vector Tutorial

Easy vector tutorial is process of tutoring to students in online by the tutors. Easy vector tutorial is the vectors which are specified by magnitude and direction. The examples of vectors are displacement, velocity, acceleration, momentum of force and weight. Easy vector tutorial is method  tutoring on vector problems in the online with basic steps and calculations. Tutor vista is the main website to provide online tutors to give easy vector tutorial.  The following problems are some of the examples of  easy vector tutorial process done by tutor vista.

Easy vector tutorial on example problems

Example problem 1: Add 2`veci` +6`vecj`+4`veck` with 5`veci`+2`vecj`+2`veck`
Solution
Place the vectors according to the magnitudes shown below.

    2`veci` + 6`vecj` +4`veck`

    5`veci` + 2`vecj` +2`veck`

   ---------------------------

    7`veci` + 8`vecj` + 6`veck`

   ----------------------------

Example problem 2: Subtract 5`veci` + 4`vecj`- 8`veck` with 4`veci`- 6`vecj` +3`veck`
Solution
Since we have the magnitudes in opposite direction it will be easy to perform method
    (5`veci` + 4`vecj`- 8`veck`4`veci`- 6`vecj` +3`veck`
    `(13 - 4)veci + (4-(-6))vecj + (-8-3)veck`
    `(13 - 4)veci + (4 + 6))vecj + (-8-3)veck`

    `(9) veci + (10)vecj + (-11) veck`

     `9 veci +10vecj -11 veck`

Example problem 3: Find the magnitude of `2 veci - vecj + 7 veck`
Solution:
Magnitude of `2veci - 3vecj + 7 veck = |2veci - 3vecj + 7 veck|`
                                                   = `sqrt((2)^2 + (-3)^2 + (7)^2)`
                                                   `sqrt(4+9+49)`
                                                  =`sqrt (62)`


Example problem 4: Find the sum of the vectors `veca - vecb + 2 vecc` and `2veca +3vecb - 4vecc` and also find the modulus of the sum.
Solution
Let  `vecx = veca - vecb + 2 vecc` ,      `vecy = 2veca + 3 vecb - 4vecc`
`vecx + vecy = (veca - vecb + 2 vecc) + (2veca + 3 vecb - 4vecc)`
             `= 3 veca +2vecb - 2 vecc`
`|vecx + vecy| = sqrt (3^2 + 2^2 + (-2)^2)`
               ` = sqrt (9+4+4)`
               ` = sqrt 17`

Easy vector tutorial on practce problems

1. Add 7`veca` +2`vecb`+3`vecc` with 2`veca`+3`vecb`+`3vecc`
Answer: 9`veca`+5`vecb`+6`vecc`
2. Subtract 4`veci` + 3`vecj`- 6`veck` with `veci`- 2`vecj` +2`veck`
Answer: `veci`+5`vecj` -8`veck`
3. Find the sum of the vectors `vecp -3 vecq + 2 vecr` and `3vecp +vecq - 2vecr` and also find the modulus of the sum.
Answer: 6

Geometric Mean

     The geometric mean is the mean or average which indicates the central tendency or typical value of a set of numbers. This is similar to arithmetic mean, which is the sum of adding the set of numbers and then dividing the sum of count number in set, n, and the number are multiplied and the nth root of resulting value is taken. The geometric mean only applies to positive numbers.

Formula Of Geometric Mean

Geometric Mean = ((X₁)(X₂)(X₃)........(X_N))^1/N
Where
              X = Individual score
              N = Sample size (Number of scores)
Geometric mean is also used for set of numbers whose values are multiplied together or exponential in nature, such as growth of growth of human population or interest rates of a financial investment. The geometric mean is one of the three classic Pythagorean means, aforementioned arithmetic mean and harmonic mean.

Calculation of Geometric Mean

 The geometric mean of data set [ a₁, a₂, --------, a_n ] is given by
^{$\left(\prod_{i = 1}^{n} a_{i}\right)^{1/n}$} = $ \sqrt[n]{a_{1}a_{2}......a_{n}}$
The geometric mean of a given set is less than or equal to the given set of arithmetic mean (the two mean of equal when all the members of a data set are equal) this allows the definition for arithmetic geometric mean the two values always lie in between the geometric mean is also arithmetic harmonic mean in that if two sequences (a_n) and (h_n) are defined:
a_{n + 1} =  $\frac{an + hn}{2}$ , a_o   = x and
h_{n + 1} = $\frac{2}{1/an + 1/bn}$, h_o = y.
Then a_n is converge to the geometric mean of x. h_n is converge to the geometric mean of y.
The sequence for converge to a common limit ( which can be shown by Bolzano - Weierstrass theorem) the geometric mean is given by
$\sqrt{a_{i}h_{i}}$  = $\sqrt{\frac{a_{i} + h{i}}{1/a_{i} + 1/h_{i}}}$  =  $ \sqrt{a_{i + 1}h_{i+1}}$

Exponential Growth of Geometric Mean

The geometric mean is more approximate than the arithmetic mean for explaining exponential growth. 

How to Calculate Square Feet

How to calculate square feet :Square feet of an area is determined by multiplying length and the width of the given area . The area of regular polygons like square , rectangle , triangle etc... can be calculated by using the formula . If the dimensions of a regular polygon is given in feet then by just substituting the given dimensions in the formula the area in square feet can be calculated , if the measurements are given in any other units such as centimeter or inches or yards then we convert these units to feet and then calculate the area in square feet , similarly for a circle also if the radius is in feet we can just substitute the radius in the area formula and calculate the area in square feet ,if the radius is given in any other unit then convert that to feet and then calculate the area in square feet .

Area of composite shapes , a composite shape is a shape which can be divided into more than one of the basic shapes . An example of a composite shape or figure is shown in the picture .

Here in this picture we can see a square , a rectangle and a triangle . Area of the whole figure can be calculated by calculating the areas of each shape and then adding them . If the measurements are not in feet then we have to convert them to feet and calculate the area in square feet .

How to calculate square feet live situations .

The home we live , the school we study and mostly all other places one or the other time needs the area of floors , walls, ceilings , windows etc. in square feet to get them painted or any other purpose and usually the areas are calculated in square feet as most of the paints or coating materials have the coverage  specification mentioned in square feet like a ' x ' brand paint covers 30 square feet per liter . 

For How to calculate square feet the example here is an interior view of a room which has to be carpeted and walls painted .( all measurements are in feet )

We can see a room with a floor plan of a 

composite figure , that can be split into two rectangles as shown.

                                                          

Area of the floor in square feet to be carpeted is calculated by first learning How to calculate square feet :

Area of the floor  =  Area of rectangle 12 x 8  +  Area of rectangle 9 x 6

Area of the floor  =  96 square feet  +  54 square feet

Area of the floor  =  150 square feet .

There are all together six rectangular shaped walls as shown below .( figures not to scale )

To find the  total wall area , find the area of each wall and add them up .

   

  Area of wall 1. = ( 10 feet x 8 feet ) - ( 7 feet x 3 feet ) [ area of door ]                                                =   80 square feet  -  21 square feet  =  59 square feet

   Area of wall 2. = 10 feet  x  12 feet  =  120 square feet

   Area of wall 3. =  ( 10 feet x 14 feet ) - ( 4 feet x 5 feet ) [ area of window ]                                       =    140 square feet  -  20 square feet  =  120 square feet

   Area of wall 4. =  10 feet  x  9 feet  =  90 square feet

Area of wall 5.  =  10 feet  x  6 feet  =  60 square feet  ,  Area of wall 6.  =  10 feet  x  3 feet  =  30 square feet 

Total area of the wall to be painted  =  59  +  120  +  120  +  90  +  60  +  30  =  479 square feet .

How to calculate square feet unit conversion examples .

How to Calculate square feet  in each diagram .



The dimensions given for this square is in inches , we have to convert the inches to feet so that we can calculate the area in square feet .
12 inches = 1 feet  so   24 inches = 2 feet
Area of square is given by  A  =  s²       where  s  =  2 feet
Area of square  A  =  ( 2 feet )²      =    4 square feet

The dimensions for this triangle are in centimeters , we have to convert centimeters to feet so that we can calculate the area of triangle in square feet .
30 cm = 1 feet    and   15cm = 0.5 feet
Area of triangle is given by  A  =  1/2 b h  ,   where base b = 0.5 feet and height h = 1 feet
Area of triangle  A  =  1/2 x  0.5 feet  x  1 feet  =  0.25 square feet


This a composite shape and we have split that into basic shapes , we have two rectangles , one triangle and a semi circle.
Area of the total shape = 4 x 5 + 4 x 9 + 1/2 x 4 x 4 + 3.14 x 2²                                                                                                                   2
Area = 20 + 36 + 8 + 6.28  =  70.28 square feet


The radius of the circle is given in meters , we have to convert the meters to feet so that we can calculate the area of circle in square feet .
1 meter  =  3 feet   so   3 meters  =  9 feet
Area of circle is given by   A  =  π r²       ,    where     =  3.14 ,  radius r  =  9 feet
Area of circle  A  =  3.14  x  9²    =  3.14   x  81  =  254.34 square feet

Students can learn How to calculate square feet from the above examples and solve problems on similar lines.

Domain all the Real Numbers

In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or "value" for each member of the domain. For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0.(Source: Wikipedia)

Domain all the real values - Example problems:

Example problem: 1
Find the domain function of f that can be defined by f (x) = -1 / ( x + 10)
Solution:
  To find the domain equate denominator to zero
  x + 10 = 0
  x = -10
Therefore the domain function of f is the set of all real values of x in the interval is(-infinity, -10) U (-10, +infinity)
Example problem: 2
 Find the domain function of f that can be defined by f(x) = x² + 6.
Solution:
 The function f(x) = x² + 6 is defined for all real values of x.
 Hence, the domain function of f(x) is the function of  "all real values of x".
 Since x² is never negative, x² + 6 is never less than 2
Example problem: 3
 Find the domain function of f that can be defined by f(t) = (1)/(t + 6)
Solution:
 The function f(t) = (1)/(t + 6) is not defined for t = -6, as this value requires division by zero.
 Hence the domain function of f(t) is the function of all the real numbers except -6
 Also, no matter how large or small t becomes, f(t) will never be equal to zero

Domain all the real values - Practice problems:

Practice problem:- 1
Find the domain of function of f that can be defined by f (x) = sqrt (-x + 16)
[Answer: Domain of function f is the set of all values of x in the interval (-infinity, 16) ]
Practice problem:- 2
Find the domain of function f defined by f (x) = sqrt( -x + 4) / [(x + 2)(x + 8)]
[Answer: domain of function f is the set of all values of x in the interval   (-infinity, -8) U    (-8, -2) U (-2, 4)] 

Properties of the Parallel Lines

Parallel lines
          The distance between the two lines will be equal and they will never intersect, such lines are said to parallel lines.

 Properties of the parallel lines
           The parallel lines will have the same slope (m1=m2)
           Here the m1 and m2 are slopes of the lines.
           When the parallel lines are cut by transversal line, then eight angles are formed

properties :

Here
                     Angle 1 = angle 3 =angle 5=angle 7
                    Angle 2= angle 4 = angle 6= angle 8.

 Adjacent angles properties:
        when  the parallel line is cut by a transversal line, then the adjacent angles sum up to 180 degrees
                   Angle 1 +angle 2= 180 degree
                   Angle 3 +angle 4= 180 degree
                   Angle 5 +angle 6= 180 degree
                   Angle 7 +angle 8= 180 degree
                Here the adjacent angles always sum up to 180 degree.

 Corresponding angles properties
       The angle in the same position around the intersection of two points is said to be corresponding angle.
          when  the parallel line is cut by a transversal line, then corresponding angles are  equal
                    Corresponding angles:
                       Angle 1 = angle 6
                       Angle 2 = angle 5
                       Angle 3 = angle 8
                       Angle 4 =angle 7

Alternate angles properties:
                   when the parallel line is cut by a transversal line then alternate angles are equal
                   So
                    Alternate interior angle
                       Angle 3 = angle 5
                       Angle 4 = angle 6
                    Alternate exterior angle
                      Angle 2 = angle 8
                      Angle 1 = angle 7


Algebra is widely used in day to day activities watch out for my forthcoming posts on hard math problems for 8th graders and cds exam 2013. I am sure they will be helpful.

Model problems:

1.Find the angle 2, when the angle 1 = 120?
Solution:
 Here angle 1 and 2 are adjacent angles
 When parallel lines is cut by a transversal line, then the    adjacent angles sum up to 180 degree
          So angle 1+angle 2 = 180
                    120 +angle 2= 180
                     Angle 2= 180-120
                    Angle 2 =60 degrees
 2.Find the angle 4, when the angle 1 = 120?
Solution:
           Here angle 3 and 4 are adjacent angles
           Angle 1= angle 3 (vertical opposite angles are equal)
           Angle 3 = 120
 When parallel lines is cut by a transversal line, then the adjacent angles sum up to 180 degree
           So angle 3+ angles 4 = 180
                       120 +angle 4= 180
                        Angle 4= 180-120
                       Angle 4 =60 degrees.