Wednesday 25 April 2012

Pythagorean theorem

Previously we have discussed about calculate the volume of a sphere where the radius is 9 meters. and In today's session we are going to discuss about Pythagorean theorem which comes under school of secondary education andhra pradesh, Pythagorean Theorem was given by the ‘Pythagoras’ a Greek mathematician. Pythagorean Theorem can be defined as the square of a hypotenuse is equal to the sum of the square of the base and square of the perpendicular that is the opposite side of the hypotenuse in a right angled triangle. If we define it in the form of expression then it will be denoted as,
(Hypotenuse) = (base) + (perpendicular) 2.
For Example: A Right angled triangle named as XYZ and if ‘Z’ is a hypotenuse of right angled triangle and ‘Y’ is base and ‘X’ denotes the perpendicular then Pythagorean theorem can be expressed as (Z) = ( X) 2+ (Y) .
We can explain it by taking an example as;
If the base = 12 inch (Y = 12 inch) and perpendicular = 5 inch (X = 3 inch) then find the length of hypotenuse ‘Z’ in meters by using the Pythagorean Theorem?
Solution: Pythagorean Theorem proof
(Hypotenuse) = (base) + (perpendicular) 2,
Then (Z) = (X) + (Y) 2,
(Z) = (12) + (5)2,
Z = √ (12) + (5)2,
Z = √ (144) + (25),
Z = √ 169,
Z = 13 inch.
So according to the calculation, the length of the hypotenuse in a right angled triangle XYZ is 13 inch.
Pythagorean Theorem is used in the case when two sides of the right triangle are given and for calculating the third side of the tight angled triangle, we are use the Pythagorean Theorem.
In the next session we are going to discuss Grade VII, Rectangular coordinate system and You can visit our website for getting information about algebra help online.

Basic constructions

Hello students, Previously we have discussed about who invented calculus and in this blog we are going to discuss basic constructions which comes under andhra pradesh education board. In basic construction generally we include the construction of similar triangles, tangents to circles and the basic constructions in geometry which include: -
-Construction of various angles such as 30°, 45°, 60°, 90o.
-Bisecting an angle.
-Constructing a line parallel to a given line through a given point.
-Construction of perpendicular bisector of a line.
-Construction of closed figures like squares, rectangles, quadrilaterals etc.
-Construction of in circle, circum circle and ex circle of a given triangle.
All of the above construction uses compass and ruler.
If we talk about more simple geometry then basic construction include: -
-Constructing of line (basic construction in geometry).
-Constructing of triangle.
-Construction of circle.
Let’s take one construction from the above list to see the basic construction in geometry.
Construction of triangle: - Triangle is a three sided close figure. Triangle can be constructed using the ruler, compass and protractor.
Some steps to construct the triangle are: -
Step 1: - A triangle has three side then XY = a units YZ = b units and ZA = c.
Step 2: - Draw the line XY = a units using ruler.
Step 3: - Take a compass and measure c units with the help of ruler.
Step 4: - With X as center, cut an arc above the line segment XY with the help of compass.
Step 5: - Again take a compass and measure b units.
Step 6: - With Y as center cut an arc above the line segment XY.
Step 7: - Both the arcs meet at a point that is Z.
Step 8: - Join XZ and YZ. Now, XYZ is the required triangle.

In the next session we are going to discuss Grade VII, Pythagorean theorem and You can visit our website for getting biology help.

Tuesday 24 April 2012

Congruence

Previously we have discussed about tangent line approximation and In today's session we are going to discuss about Congruence which is a part of school boards in india , In geometry if we are given two figures and the question is asked that are the two geometrical figures congruent, for this we need to recall the congruence definition. According to Congruence, two or more given geometrical figures are congruent if they have all the sides of the equal measurement and the angles formed by all the line segments are exactly equal. We say the two figures are congruent, if the two figures are placed one over another, they overlap each other.
Let us first take two circles, the two circles are said to be congruent, if they are drawn with the same radius.
 In case of the square, we say that the two squares are congruent, if they are formed with the help of the same line segment. As we know that all the angles of squares are equal to 90 degrees each, so we say that the two squares are congruent if they have the length of each side of the same measure.
Now we take the two rectangles, the two rectangles are said to be congruent, if the length of one rectangle is equal to the length of another rectangle and the breadth of one rectangle is equal to the breadth of another rectangle. All the rectangles have all the angles equal to 90 degrees. If we place one rectangle over another, the two rectangles are found to be exactly of the same measure and they overlap each other.
 In case of the triangles, we know that the triangles are formed by joining 3 line segments. Here we say that the two triangles are congruent, if the three sides of the lines are congruent   and the corresponding sides of the triangle are also equal, then the triangles are said to be congruent.
 In the next session we are going to discuss Basic constructions and You can visit our website for getting information about chemistry answers.

Reflections/translations on a coordinate plane

Hello students in mathematics we study the concept of the transformation. There are many types of Transformation schemes that are used in various fields. Some transformations are: -
-Rotation
-Reflection
-Translation
The transformation can be stated as the word that means changing in shape of the object, in this a shape of object can be change, and position of object can be change.
-Reflections on a coordinate plane mean reflection on the xy axis or xy coordinate or xy plane. The x coordinate represents the horizontal position and y coordinate represents the vertical position.
Reflection of any object means reverse image of that particular object or flipping of the object. We see our image in the mirror is the example of Reflection and reflection on a coordinate plane means a particular object that are on xy plane will reflect on the -x-y plane.
-Translations on a coordinate plane mean shifting of one shape from xy plane to another plane. In it only changes in position not in shape. In the translation transformation the object or figure is shifted only one position to another and their shape remain unchanged. In the translation we can move the shape into any direction like in upward, downward, on the right side and left and wherever we want. (know more about cbse board, here)

The reflexion and translation can be more understand by graphically. The shape may be triangle, rectangle and anything for both the transformation. Reflected and translated figures are represented by the ' symbol like we have original figure name is ABCD then the transformed figure will be represented as the ABCD'. To solve both we have to provide the numeric data so that we can draw the any shape on a coordinate plane for transforming.


In the next session we will discuss about Congruence

Monday 16 April 2012

Tessellations


Hello friends today we will discuss about tessellation which you need to study in grade VII. Generally in mathematics you didn't heard this term but it has a relation with mathematics. A tessellation is a pattern made by repeating shapes. Making Tessellations require a creativity of an art with capability of solving puzzles. In this world there are many natural tessellations also present.
In modern world very few people are aware of the term Tessellations. The connection between math and art is very strong and frequent but very few people are aware of that. Tessellation also covers a regular space without overlapping and without leaving any space. For example a chessboard is a Tessellation, made of squares with no gap in between and without overlapping. The pattern of the brick on a wall is a Tessellation made by rectangle.
Whenever we deal in mathematics we aware of terms like algebra, calculus, trigonometry ...etc. if we talk about the term tessellation we are not aware of this term. But as we know that there is a very strong relation between art and mathematics and that is only because of the term like tessellation, so we can say that Tessellations has a relationship with mathematics. The tessellation definition is that shape of repeated things. (know more about cbse latest sample papers, here) The repeated things can be anything, if we take an example of a chessboard then it is a pattern made by squares as squares are repeated regularly in the chessboard but the thing we need to notice is that there is no gap between the square, so if we are drawing any tessellation then the thing we need notice that there is no overlapping between the figure and there is no gap between the figure. In the next session we will discuss about Reflections/translations on a coordinate plane

Complementary/supplementary angles

In this unit we are going to learn about Complementary angles, supplementary angles. We say that the angle is formed, if the two rays goes in the different directions and have the same starting point. This starting point is called the vertex of the angle.  Now we will see what complementary angles are and what are supplementary angles? Moreover we will study that what is the complement of the given angle and what is the supplement of the given angle. (know more about cbse class 12 board papers, here)
We mean, by the term Complementary angles, we mean that the   sum of two angles is equal to 90 degree. If the measure of one angle is given say x and we need to find the Complement of the given angle then we say that the complement of the given angle is 90 – x.  Thus if the two angles are complementary, it means that their sum is 90 degrees and so it forms a right angle. The two adjacent angles if joined together if form a right angle, and then we say the two angles are complementary.
We mean, By the term supplementary angles, we mean that the   sum of two angles is equal to 180 degree. If the measure of one angle is given say x and we need to find the supplement of the given angle then we say that the supplement of the given angle is 180 – x. Thus if the two angles are supplementary, it means that their sum is 180 degrees and so it forms a straight angle.  We also observe that if the two angles are supplementary, then their sum is 180 degrees and so we can say that the two angles form a linear pair. In the next session we will discuss about Tessellations

Properties of 2-d and 3-d figures

Hello students. In this session we are going to discuss ob the topic of Properties of 2 d and Properties of 3 d figures. But before starting it you should be familiar with the terms of 2D and 3D. 2D is the shapes that can be drawn on the plane paper and 3d shapes can not be and to be more precise 2d shapes is 2d dimension and 3d shapes is 3 dimensions. We can not handle the 2d shapes in our hand because they are the flat shapes but we can handle the 3d shapes.
The most important difference between them is that a 3d shapes have three axises such as x, y and z whereas 2d shapes have just two axises that is x and y.   (know more about cbse sample papers, here)
Let’s talk about the properties or dimension of 2 d and 3 d figures.
2d figures has two properties or dimensions such as length and width and 3d figures has three properties or dimensions such as length, width and depth.
Circle, triangle, square, rectangle and any polygons follow the 2d properties, because they are 2d shapes.
Sphere, prism, cuboids, cube, cylinder, pyramid and cone follow the 3d properties because they are 3d shapes.
Although every shape of 2d and 3d have different properties for example: -
Some 2d Shapes                                  Properties
Square                                                 4 sides, closed figure, 2 sets of parallel line
Triangle                                                3 sides, closed figure
Rhombus                                              2 sets of parallel line, polygon, 4 sides of equal measure

Some 3d Shapes                                  Properties
Cube                                                    6 congruent faces, 12 vertices,
Cylinder                                               2 circle faces
Rectangular prism                                 6 faces, 8 vertices
Although we have so many 2d and 3d figures and shapes, but here it is not possible to discuss all of them.
I hope that make sense.
In the next session we will discuss about Complementary/supplementary angles

Regular/irregular geometric shapes

In the geometrical mathematics, we all are very well aware that it is a collection of several kinds of shapes and figures. These figures and shapes are very helpful in solving various kinds of problem that are related to our daily routine life. In geometry, these shapes and figures are categorized into different category according to their properties like quadrilateral, circle, and polygons and so on. Here we are going to discussing about the geometrical shape 'polygon'. The concept of polygon helps the school's student to understand the geometrical concept very well. (know more about cbse latest sample papers , here)
Polygon is a geometrical shape which comes from the Greek word. Polygon word formed by the combination of two words, 'poly' which mean is “many” and 'gon' which mean is “angle”. A polygon is a shape that is drawn on the 2-D plane with help of straight lines or sides. In the simple mean we can say that polygon is the two dimensional shapes. These shapes are formed by straight lines and the shape is closed. It means that in this shape all the lines are connect to each other. In the polygon, according to the property of geometrical shape we can categorize the polygon into the following manner:
A) simplex or complex
B) Concave or convex
C)
Regular geometric shapes or irregular geometric shapes
Here we are going to discussing about the regular and irregular shape.
Regular geometric shapes are those shapes in which all the existing angles are equal and all the sides are equal. In the same aspect we can say that those figures whose existing angles and shapes are not equal in measure then these shapes are consider as a
Irregular geometric shapes. In the case of regular, equilateral triangle is the one of the best example because this triangle is made up of three straight sides, it is closed in shape and all the angles and sides are equal to each other. In the next session we are going to discuss Grade VII, Geometric concepts and In the next session we will discuss about Properties of 2-d and 3-d figures

Geometric concepts

As we all are very well aware that geometry is the one of the oldest branch of mathematics. It plays an important role in mathematics to solve various kinds of problem. The term “geometry” is comes from the Greek word “Geometron” which is formed by the combination of two word “Geo” and “Metron”. The word “Geo” refers to Earth and the word 'Metron' refers to measurement. The need of the  Geometric concepts was felt in several years ago when person realizes to measure their land when they want to sold or buy their land. The basic application of geometrical construction were made centuries before the mathematical principles on which construction were based and recorded.
In the basic concept of geometry we generally studied about the points, lines, planes, closed flat shapes and many others. These are the most basic concept of geometry that provides the helps in describing, designing and constructing any type of visible objects. In todays life, the concept of geometry used in various field like architects, engineers field, planning for constructing the buildings, bridges roads and many other geometrical architecture cal things. In the below we show you the some of the basic
Concepts of geometry:
A) Point: In geometry a point can be defined as a mark of point which has no length, no breath and no height. It occupies the position and location but no magnitude. This point helps in creating various other type of geometrical shape.
B) Line: The line can be defined as a distance between the two points. A line has no width. Usually a line refers as a straight line. (know more about icse board, here)
C) Line – segment: This line segment different form the line. In line segment, the part of a line with two definite end points is defined but in line these definite points are not define. These line segments are very helpful to form various geometrical shapes like square, rectangle and so on.
As like above there are other geometrical concept are defining in mathematics. Like interesting lines, parallel lines, perpendicular lines and so on. In the next session we are going to discuss Grade VII, Tessellations.

Properties of lines

In this session we are going to learn about Properties of lines To study about the  properties of lines and angles we first talk about line as the group of endless points which are collinear and extends endlessly in both the directions. The lines do not have any fixed length and has arrows at both the ends, which indicates that the line is extending in both sides.
Here are some of the properties of the lines and angles:
If we have a pair of lines which intersect each other at only one point, then we say that the lines are intersecting at a point. If the two lines are intersecting then the pair of opposite angles so formed are called  vertical opposite angles. We must remember that the pair of  vertical opposite angles are  equal. Another pair of  lines we talk about are  parallel lines, the pair of lines are called parallel, when we find that the pair of lines  do not meet at all , or it is said intersecting lines  at infinite which we cannot  see at all.  Thus we say that the lines are parallel then the  perpendicular distance between  the two lines at all the points are equal. (know more about cbse sample papers, here)

Next type of lines are called coinciding  lines, when we have two lines on the plane, such that its all points are coinciding so that when the lines are plotted, the two lines overlap each other. Two lines are called perpendicular to each other, when we find that the pair of lines form an angle of 90 degrees  between each other, then the  lines are called perpendicular to each other. Remember that the line segment is the part of the line which has a fixed length and  so it has two end points. In the next session we will discuss about Geometric concepts