Distance of a Point from a Plane

Let us see about distance of a point from a plane,
In mathematics, a plane is placed at distance from a point of any horizontal, two-dimensional smooth surface. A plane is the two dimensional have a distance of a point that means zero-dimensions, a line that is known as one-dimension and a space  is known as three-dimensions. A lot of mathematics may be performed in the surface of plane, particularly in the areas of geometry, trigonometry, graph theory and graphing.

Properties of plane:

• Two planes may be parallel or both intersect in a line.
• A line may be parallel to a plane, intersects in point.
• Two lines at a 90 degree angle to the same plane should be parallel to each other.

• Two planes at a 90 degree angle to the similar line should be parallel to each other.

Point - Plane distance:

Let us see about distance of point from plane:
Given a plane,


and a point X0 = (xo,y0,z0),the normal to the plane is given by,

and a vector from the plane to the point is given by,

Projecting W onto V gives the distance D from the point to the plane as,

This equation is known as the distance of a point from a plane.

Examples:

1)      Find the distance from the point P = (2, 2, 4) for the plane 2x +2y + 3z + 4 = 0.
Solution:
We use formula from the distance of a point from a plane.
From the above equation we substitute for the plane A = 2, B = 2, C = 3, D = 4. From the point P, we substitute x₁ = 2, y₁ = 2, and z₁ = 4.
Plane's distance of a point is,


2) Find the distance from the point P = (2, 3, 5) for the plane x - y + z + 5 = 0.
Solution:
We use formula from the distance of a point from a plane.
From the above equation we substitute for the plane A = 1, B =- 1, C = 1, D = 5. From the point P, we substitute x₁ = 2, y₁ = 3, and z₁ = 5.
Plane's distance of a point is,

Frequency Distribution Histograms

A frequency distribution can be defined as the number of observations falling into each of several ranges of values. It can be represented as frequency tables, histograms. Either the percentage of observations or the actual number of observations falling in each range can be represented by Frequency Distribution. If the Frequency distribution shows the percentage of observation then it is said to be Relative frequency distribution.

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In statistics Frequency distribution plays a huge role and it can be represented either in a tabular or graphical format and it displays the number of observation within a given interval.

A frequency distribution can be represented by a histogram or pie chart. In case of large data sets, the stepped graph is approximated by the smooth curve of a distribution function which is known as a density function.

Frequency distribution mostly used for assigning probabilities and summarizing large data sets.
It is a method of representing unorganized data e.g. to show results of an income of people for a certain region, election, sales of a good within a certain period, etc.

Frequency distribution in histograms

Histograms:
The histogram is nothing but a summary graph representing a count of the data’s falling in various ranges. The groups of data are known as classes; here in histogram they are called as classes.
       
In mathematical sense, generally a histogram can be defined as a function X_i that counts the number of observations or data’s that fall into each of the bins. The graph of a histogram is similar to the one way to represent a histogram.

If,
n = number of observations and
k = total number of classes then, the histogram meets the following criteria.
                n= `sum_(i=1)^k` X_i

Histograms can be defined as follows,

• In statistics, it is a pictorial representation of table frequencies, shown as bars. The diagram shows what proportion of cases fall into each of several categories. Simply Histogram is a form of data binning.
• Histogram is a bar chart representing the frequency distribution of values along a spectrum of possible values.
• Histograms are mainly used for density estimation.

Sample Problems

Univariate frequency tables:
The lists ordered by quantity which shows the number of times each value appears is known as Univariate frequency distributions.

Example 1:
If 200 people rate a product assessing their agreement with a statement on the product on which ‘A’ indicates strong agreement and ‘E’ strong disagreement, the frequency distribution of their responses can be given as follows:

Number	Degree of agreement	Rank
60	Strongly agree	A
40	Agree somewhat	B
30	Not sure	C
40	Disagree some what	D
30	Strongly Disagree	E

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Example 2:
The weight of the students in a class could be represented into the following frequency table:

Weight range (Kg's)	Number of students	Cumulative number
40 -50	25	25
50-55	35	60
55-60	15	75
60-65	24	99
65-70	10	109

 

How Draw an Ellipse

In geometry, an ellipse (from Greek ellipsis, a "falling short") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is perpendicular to the axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant. (Source : Wikipedia)

Draw an ellipse using string and 2 pins

Constructions – instruments defined as Euclidean structure is not a correct and ellipse is used to draw the ways of laws, significant is not a accuracy of the arithmetical, but if you are careful, you can closed to fairly. It is a little bits called the "Gardener's Ellipse", because scale of great is well effort to using a cable and stakes, to lay out elliptical flower beds in proper gardens.

How to draw ellipses

Consider an ellipse, in two positions, each one called a focus. The position F1 and F2 are shown in diagrams. The calculation of the distance to the centre point is stable, If you get any position on the ellipse. Around the ellipse position is drag and observes point in every distance to the focus of the different point and their calculation is stable. The sum of the double distances is resolute by the size of the ellipse. The major axis of the duration is same to the sum of these distances.

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Properties of an ellipse: How to draw an ellipse

Centre:
The ellipse of the point which is medium of the row segment between the two foci of the axes. The connecting of the major and minor axes.


Major / minor axis:
The diameters of the ellipse are greatest and shortest. Both the measure of the major and minor axis of the generator lines is to be same.



Semi-major / semi:
The distance from the middle to the extreme and nearby position on the minor axis of ellipse.


Foci /Focus points:
The two points of the position is defining the ellipse.

Real Size Ruler

Let us know about real size ruler.Ruler is an instrument, using this instrument we can draw the line, those lines are straight and also very neat, size of the ruler depends on the separation (they are separated in cm and inches).In real life we are using the ruler in many places.Type s of rulers are

• Desk ruler
• Practical ruler.

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Desk ruler-real size ruler:

Desk ruler is used for three purposes and size of the ruler is12 inches or 30 cm

• Drawing
• The diagram should be straight and
• Cutting the things into straight line

Practical ruler –real size ruler:
Practical  ruler is used for measuring the distance,  all the carpenter use this kind of ruler The lenght of the ruler is 5 meter or 2 meter(metal tape measure)
Real size ruler diagram:
The diagram of the ruler is shown below, (separation in both inches and centimeter)
 

Using the ruler we can draw the line and also measure the line

Example problem-real size ruler

Example 1:
Draw the 5cm line using the real size desk ruler?
Solution:
The above diagram shows the how the desk ruler look like, we can construct desk ruler in both the plastic and wood. Using this ruler we can draw the line with 5cm



Mark 0(starting point to the ruler) and mark 5 on ruler (ending point),then draw the straight line .we get line as 5 cm length.

 Example 2:
Draw 11cm using the real size desk ruler?
Solution:
Desk ruler has the value from 0 to 20 , using the desk ruler we can draw a straight line ,mark 0 on the ruler (we call as starting point ) and mark 11 on the ruler (we call as ending point ) then connect those two points we get the straight line with the measure of 11cm.

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Rulers in different units

Rulers come in different units like:

• Fractional Inch
• Decimal Inch
• Metric

Metric rulers are fairly easy to read. They deal with centimeters and millimeters only.

Steps:

1. Decide if we have a metric or an English ruler. Metric rulers have numbers every centimeter, and English have them every inch. Centimeters are much smaller than inches, so metric rulers will have a lot more numbers printed on them. If we still have trouble telling what type of ruler you have, centimeters are normally about as wide as one fingertip, and inches are about as wide as two.
2. Read an English ruler using fractions of an inch. The distance between any two large numbered lines is 1 inch. The large unnumbered line that is halfway between them is 1/2 inch. The smaller (but still prominent) line between the 1/2 mark and the numbered inch line is 1/4 inch. The tiny little lines between all of the more prominent lines are 1/16 inch.
3. Observe the much simpler metric rulers. The distance between any two large numbered lines is 1 cm. The prominent line between any two numbered lines is 1/2 cm. The small lines between the 1/2 mark and the numbered centimeter mark are 1/10 cm, otherwise known as a millimeter.
4. Record distances by the name of the line that it most closely matches. If the length of an object goes to one mark past the halfway mark on your ruler then it will be 9/16 inch on an English ruler or 6/10 cm (or 6 mm) on a metric ruler.

Eccentricity of 1

Conic sections:
Conic sections are formed when a right cone is intercepted by a plane.The shape so formed depends on the angle at which the conic is cut.
Consider the below figure of right cone intercepted by plane and we can see the different shapes formed.



Conic sections:
Conic sections are formed when a right cone is intercepted by a plane.The shape so formed depends on the angle at which the conic is cut.
Consider the below figure of right cone intercepted by plane and we can see the different shapes formed.



The first picture formed is known as parabola
Second one is ellipse
The third is hyperbola.

explanation to eccentricity of 1

 Eccentricity of 1  :-
Eccentricity is the measure of the deviation of a conic from a circular path .The parabola is the conic haing eccentricity as 1.

Definition of parabola :-
Parabola is locus of points whose distance from fixed line, called directrix, and a from a fixed point ,called focus,is equal. Vertex is a point where parabola changes its direction .The distance from focus to vertex and vertex to directrix is equal.

The main compenents of Parabola are :-
i)Vertex
ii) axis
iii) focus
iv) directrix
Consider the parabola given below

Description  of paraboala :-
i)Point F is called the focus of parabola
ii)Point "O" is called as the vertex of parabola.
iii)C ,D are the points on the parabola whoce distance fron focus and directrix are equal .
iv) Line AB is called directric fo parabola
v) Line ox is called the axis of parabola.

i) Y² = 4aX                                                                      

This parabola opens towards right side ie + ve x axis having  vertex at  (0,0). and focal distance "a" and focal point (a,0) . This curve transforms to (Y-k)² = 4a(X-h) when vertex shifted to a point  (h,k). with focus shifting to (a+h ,k)

ii) Y²= -4aX

This parabola opens towards  left side ie -ve  x axis having  vertex at  (0,0). and focal distance "a" and focal point (-a,0).This curve transforms to (Y-k)² = - 4a(X-h) when vertex shifted to a point  (h,k) and focus shifts to (h-a ,k)

iii) X²= 4aY
This parabola opens towards  top  side ie +ve  y axis having  vertex at  (0,0). and focal distance "a"and focal point (0,a).This curve transforms to (X-h)² = 4a(Y-k) when vertex shifted to a point  (h,k) and focus shifts to (h,k+a).

iv) X²=-4aY
This parabola opens towards down side ie -ve  y axis having  vertex at  (0,0). and focal distance "a" and focal point (0,-a).This curve transforms to (X-h)² = - 4a(Y-k) when vertex shifted to a point  (h,k-a).

examples to eccentricity of 1

Ex 1:
Given the equation of parabola Y² = 8X find the focus of parabola
solution:
Comparing with standard form of equation
Y² = 4aX
we get  4a =8
=> a=2
so focus of parabola is (a,0) = (2,0)

Ex 2:
find focus of parabola   y =  1/4 x²
solution :
y =  1/4 x² 
=> x²= 4y
=> a= 1 on comparing with standard form
so focus of parabola is (0,1) as axis of symmetry is +ve y axis

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Ex 3:
Given a parabolic equation Y²= -16X-32
Find i)  Vertex of parabola
ii) Focus of parabola

Solution:
Given Y²= -16X-32
=>  Y²= -16(X+2)
=>  Comparing with standard equation (Y-k)² = - 4a(X-h)
(h, k) =(-2,0)
a=4
i) vertex of parabola is (h,k) =(-2,0)
ii)focus of parabola is (-6,0) as parabola is opening towards left side.

Draw isosceles Triangle

The triangle is a closed geometric shape that contains three sides. There are several types of triangles in the geometry world. We can divide the triangles according to their angles and sides. We can define the isosceles triangle by using its sides.

      The triangle that has two equal or congruent sides in its measure is called as isosceles triangle. We can say that the equilateral triangle is also as an isosceles triangle,since the equilateral triangle has three equal sides . while we draw a isoscles triangle we have to see that two sides triangle must be equal.

draw Isosceles Triangle:

• when we draw a isoceles triangle, angles are congruent i.e., an angles of each sides are equal.
• when we draw a isoceles triangle diagonals of an isosceles triangle must be congruent.
• The measurement of the adjacent angles is giving the 180 degree. x = y = 180^o

Formulas for Isosceles triangle:

• Hight,  h = √(b² - `(1/4)` a²)
• Perimeter of Isosceles Triangle = A + B + C
• Area of isosceles triangle A = (b* h)/2

Example problems:

1) Find the area of isosceles triangle with the base and height are 5 cm and 7 cm.

solution:

We can find the area of given problem using the following formula:

Area A= (b*h)/2

Substitute the values of b and h into the above formula. Then we get,

         A= (5*7)/2

 Here multiplying to the values of 5 and 6 then dividing by 2.

  Then we get the final solution.   

 Answer A=17.5 cm²

2) Find the perimeter of Isosceles triangle that has side, Side 1 =10 cm, Side 2 = 10, Side 3 = 7 cm.

Solution:

   Given, Side 1 =10 cm, Side 2 = 10, Side 3 = 7 cm.

  Perimeter of Isosceles triangle   = Side 1+ Side 2+ Side 3

                                                         = 10 + 10 + 7

  Perimeter of Isosceles triangle  = 27 cm

3) Find the height of the isosceles triangle of the base a = 7 cm and equal sides b = 11 cm.

Solution:

The height of the isosceles triangle is given by the formula:

             h = √(b² - `(1/4)` a²)

substitute the a = 7cm and b = 11 cm. in the formula,

             h = √(11² - (1/4)*7² )

             h = √(121 - `(49/4)` )

             h = √(121 - 12.25 )

             h = √(108.75) = 10.428 ≈ 10.4 cm

Height of the isosceles triangle = 10.4 cm

Straight Triangle

In geometry, a triangle is the fundamental shapes. The polygon with 3 corners and 3 sides are the line segments. The corner otherwise called as the vertices. The sides otherwise called as the edges. A triangle with corners A, B and C is represented by ∆ABC.Here we will see about the triangle types and formulas.

Classification of Straight Triangles

The straight triangle has three types. The names are as follows,

• Equilateral Triangle
• Isosceles Triangle
• Scalene triangle

Equilateral triangle
     Here all three sides are the equal length. An equilateral triangle is also a normal polygon with every angles calculate 60 degree. The equilateral triangle is as follows,
               
Isosceles triangle
     These triangle two sides only the equal length. The figure is as follows,
                 
Scalene triangle
Here all sides are the different type. The three angels are also various in measures. This triangle figure is as follows,

           

Classified Based by the Internal Angels

• A triangle does not have an angle that calculates 90 degree is known as the oblique triangles.
• It has all interior angles calculating less than 90 is known as the acute triangle or acute-angled triangle.
• A straight triangle has one angle that calculates more than 90 degree is known as obtuse triangle or obtuse-angled triangle.

Formula
Triangle area is A=(1/2).b.h
Base is denoted by b and height is denoted by h

Formula for angles
     Sin(q)=opposite/Hypotenuse
     Cos(q)=Adjacent/Hypotenuse
     Tan(q)=Opposite/Adjacent

Properties

• Angle sum property
• Exterior angle property
• Triangle Inequality property
• Pythagoras theorem

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Examples

1)Find the area of straight triangle with the base 5 cm and the height 7cm.
Solution
     Given b=5 cm and h=7 cm.
     Formula A=(1/2).b.h
                    =(1/2).5 cm.7cm
                    =(1/2).35cm²
                    =17.5cm²

2)what is the obtuse triangle with the base 4 inches and the height 6 inches.
Solution
     Given b=4 in and height h=6 in
     Formula A=(1/2).b.h
                    =(1/2).(4 in).(6 in)
                    =(1/2).24 in²
                    =12in²