Understand Pythagorean triples

In Geometry Tutoring we know that a natural number, n , is said to be the perfect square of the natural number, a, if n = a*a . In other words, we can say that a natural number is called a perfect square if it is the square of some other natural number. By the term square of a number, we mean that the product of a number multiplied by itself.
Then we have also learnt some properties of perfect squares. Adding to such list of properties, we will learn one important property of perfect squares here. This property of perfect squares is called the Pythagorean triples. Let’s define the term first.
Three natural numbers m, n, p are said to to form a Pythagorean triples if m  Ì‚2 + n  Ì‚2 = p  Ì‚2
The Pythagorean triples formula comes from the Phythagorean Theorem of right triangles in which the square of the length of hypotenuse is equal to the sum of the squares of the other two sides. In Pythagoras theorem, given a, b to be the two sides of a right triangle & h as the hypotenuse of the triangle, we have the relation h  Ì‚2 = a Ì‚2 + b Ì‚2. According to Pythagorean triples formula , if m is a natural  number greater than 1 , i.e., m>1, we can find a Pythagorean triples using the formula ,
( 2m , m  Ì‚2 – 1 , m  Ì‚2 +1 )
Lets take an example from Andhra Pradesh Board sample papers

Putting m = 4 , we get the Pythagorean triples as
2m = 2 * 4 = 8
m  Ì‚2 – 1 = 16 – 1 = 15
m  Ì‚2 + 1 = 16 + 1 = 17
Thus , the numbers 8, 15, 17 form the Pythagorean triples. For more information read here

This was all about Pythagorean triples. Visit our blogs for more information on Perpendicular Linesplanes and pythagorean theorem in grade vii

Properties of inscribed and circumscribed polygons of circles

Hello students, in this blog we are going to discuss about the inscribed polygon and circumscribed polygon of circle. First of all we define about the inscribed polygon that is define as polygon (also see What is a Regular Polygon) that is inscribed in a circle if all the vertices of polygon are points on the circle and circle included all the sides of polygon then it is known as inscribed circle .We can define some Properties of inscribed polygons of circles that are as follows:
Property no (1) : All regular polygons inscribed in the circle .
Property no (2) : Both inscribed polygon and circumscribed circle have the center .
Property no (3) : If we talk about the radius of inscribed circle it is also same as the radius of circumscribed circle .
When we talk about the circumscribed circle of a polygon it is define as a circle that passes through all the vertices of the polygon .We can define some Properties of circumscribed polygons of circles as follows:
Property no (1) : Center of the circle is known as circumcenter .
Property no (2) : The radius of the circle is known as circumradius .
Property no (3) : When a polygon has circumscribed circle mean that polygon is a cyclic polygon also defined as co-cyclic polygon .
When there is triangle that is also a polygon have the sides a, b and c then for inscribed circle radius r will be define as r = Ö (p – a) ( p – b) ( p – c) / p .
here p = (a + b + c) / 2 .
And also for circumscribed circle radius R is defined as
R = a * b * c / 4 Ö p ( p- a) ( p – b) ( p -c ) .

In upcoming posts we will discuss about Understand Pythagorean triples and Learn Parabolic Functions and Axis of Symmetry. Visit our website for information on syllabus of West Bengal board of higher secondary education

perpendicular lines/planes

Hello students, in this blog we are going to discuss the Perpendicular Lines Definition and planes. But before starting you should be known to the means of perpendicular. Perpendicular means anything at right angles or 90 degree.
The perpendicular lines are intersecting to each other at the right angle or 90 degree. A line has no ends and no thickness.
Another definition of perpendicular lines are, if any two line intersecting each other and forming four equal angles than it is also said to be perpendicular lines and all four angles will be equal to 90 degrees in the case of two perpendicular lines.
Some examples of perpendicular lines are : -
-We draw the graph paper, so on the graph paper, the x axis and y axis are the perpendicular lines.
-The lowest and larger axis of ellipse is also perpendicular.
Note : - In a plane, the perpendicular lines are opposite reciprocal slopes that means the slopes' products is -1. And two perpendicular lines (get more detail here) can be shown by the symbol that is reverse of T. we can make the perpendicular lines by using compass and straightedge. If any line is perpendicular to two or more lines then it will also be perpendicular to the plane. If any line is perpendicular to a plane, than every plane that having the same line will also perpendicular to that plane. We can also find the length of Perpendicular lines.
Whereas the perpendicular planes are the planes in which a plane have a perpendicular line to the other plane and a plane is a flat surface and it also has no thickness.

The perpendicular lines and perpendicular planes can be understand more, if we will see them graphically.

In upcoming posts we will discuss about Properties of inscribed and circumscribed polygons of circles and Learn Cross Sections and Planes. Visit our website for information on West Bengal board of primary education

translations

Hi friends, today in free math answers session I am going to tell you about the transformations of coordinates in which translation is one of the process of transformation that is used for change the shape or size or orientation of the given figure but first of all we have to know about the meaning of transformation that is defined as in terms of the definition of transformation that it is the way of changing the shape or appearance of the figure that is given.So when we talk about the translation that is also the part of the transformation is defined as follows:
Translations is one more method of transformation in which thing or figure that is given for transforming will only move without rotation or resizing .If we translate the thing that means all point of the figure will be on same distance and in same direction when we move a thing on its own place then it will not change its position and size .Translation transformation process is that all the points of an object will move only in a straight line and also the direction of moving is same that means the shape or size and also orientation of the object are same as the original object or thing .If we talk about the same orientation that means object and image of object are facing the same direction .
If we want to understand the translation in simple words then it will be defined as thing or object will move from one location to new location without any changes in the shape, size and orientation. If we talk about the translations in geometry it is simply define as the figure slide somewhere else means location of the figure will change only but during the movement do not change the figure in any other way means do not resize and rotate or flip it over.

In upcoming posts we will discuss about perpendicular lines/planes and Probability and Statistics in Grade XI. Visit our website for information on Andhra Pradesh school textbooks online

isosceles triangle theorem

Today in our free geometry help session we are going to discuss about isosceles triangle theorem.  Before it we have to define the isosceles triangle that is a triangle with two congruent sides.
There are several theorems based on the isosceles triangle that are as follows:
Theorem 1 : When two sides of the triangle are congruent then angles that are opposite to each other are also congruent .We can define the isosceles triangle theorem proof as follows :
As if there is a triangle XYZ then according to the theorem if XY ≅ XZ then angle Y ≅ angle z .
Theorem 2 : If in a triangle two angles are congruent then the sides of the triangle that are opposite are also congruent .It is also known as the converse theorem .we can also define the isosceles triangle theorem proof as follows :
As if there is a triangle XYZ then according to the theorem if angle Y ≅ angle z then XY ≅ XZ .
We can also create two congruent triangles by drawing the altitude in an isosceles triangle it is also proved by Hypotenuse – leg .
When we create two isosceles triangle then congruent legs of isosceles triangle change into the congruent hypotenuse and the altitude change into the shared leg.
It have some true statements regarding to a isosceles triangle as follows (find more details here):

1.     : If we create an altitude to the base of an isosceles triangle then it will bisects the vertex .it

is shown as If there is an isosceles triangle XYZ and an altitude ZA then

angle XZA ≅angle YZA

2.     : In an isosceles triangle altitude to base bisects the base .In this statement if triangle XYZ isosceles and an altitude is ZA then side XA ≅ZA .

 In upcoming posts we will discuss about translations and Math Blog on Types of events. Visit our website for information on Andhra Pradesh geography questions

Planar cross-sections

In this free algebra problem solver session we are going to discuss about the term Planar cross-sections that is used in geometry when a figure is intersected by some plane and that figure must be a solid figure and the result of intersection may be a point or line or line segment or also a plane that is such for circle or a polygon .
A planner cross section is drawn interactively that is interpolate the data along a single line usually .Planar cross-sections is usually done with the gridded data but sometimes it is also possible to do it (Planar cross-sections) with the irregularly-spaced data .If we define the gridded data it will be a model output or gridded radar volumes .
For control of the planar cross sections, configurations containing the planner cross section always contain a horizontal window. So for generate the Planar cross-sections we have to do some steps as defined:
Step no (1) : First of all move the pointer into the horizontal window and end to where you would like to be cross sections plane .
Step no (2) : When Planar cross-sections is done with the help of the computer then push and hold the middle button of mouse and pointer will move gradually and connect the line with its original location .
Step no (3) : until the desired cross section plane will not obtained drag it .
Step no (4): At last release the mouse button.
There are several planes define into the geometry ,So if a plane is parallel to the base of the solid , then plane figure is formed similar to the original or congruent to the solid base .For defining the Planar cross-sections you have to be analyze the shape and size of every geometric figure .

In upcoming posts we will discuss about isosceles triangle theorem and Measures of dispersions in Grade XI. Visit our website for information on CBSE class 12 chemistry previous years question papers

Special right triangles

Hi Friends! In this answer math problems for free session we will discuss about Special right triangle. A special right triangle is a right triangle having specific characteristics which make its calculations much easier or for which simple formulas exist. We can classify special right triangles in two categories: the angles of a right triangle may form some simple relationships, such as 45–45–90. The right triangles in this category are “angle-based" right triangle. Another category is on the basis of sides of the triangle, when the lengths of the sides form some ratios, and then those triangles are called “side-based" right triangles.
There are four types of special right triangles:
These two types come under side based special right triangles:
3–4–5 Triangles:
A 3-4-5 triangle is a special right triangle whose lengths of the sides are in the ratio of 3:4:5. The ratios can only be checked when you are given the lengths of any two sides of the triangle.
5–12–13 Triangle:
A 5-12-13 triangle is a special right triangle whose lengths of the sides are in the ratio of 5:12:13.
These two types come under angle based special right triangle (get more detail here):
45º45º90º right triangles or isosceles right triangle: An isosceles right triangle is a triangle having the characteristic of both isosceles and the right triangle as it has two equal angles, two equal sides, and one right angle.  So the angles of an isosceles right triangle are 45°- 45°- 90°. Therefore the lengths of the sides of a 45°- 45°- 90° triangle are in the ratio of 1:1:√2.
A right angled triangle having two sides equal in length is called 45°- 45°- 90° triangle.
30º-60º-90º Triangles
A 30°- 60°- 90° right triangle is a special right triangle whose angles are 30°, 60°and 90° and therefore the lengths of the sides of a 30°- 60°- 90° triangle are in the ratio of 1:√3:2 .

In upcoming posts we will discuss about Planar cross-sections and Measures of central tendency in Grade XI. Visit our website for information on CBSE class 12 home science question bank

Rotations

In this answers to math problems session we are going to discuss about the different types of transformations of which one of is rotation but first of all we have to know about the meaning of transformation. According to one of the online tutor definition of transformation it is the way of changing the shape or appearance of the figure means how the size and shape effected of the figure when some transformation methods applied on that figure.
There are some ways for math transformations in which rotations is one of the method of transformation that is defined as follows .Rotation definition is any shape that means turning around the center , But there will be one thing is that at any point if we calculate the distance from the center is the same and also it can be understand as that each point will makes the circle around itself and that point is the center of that circle that is equal for every point of its circumference .It is the method of transformation through which the shape of the thing will not change but its orientation is changed . Useful resource for more information on rotations.
Rotation is the method in which the thing will be rotated at any of the point on that shape and the shape will not change for the point of rotation means only the orientation of the figure is changed or its side will be changed not its size .It is only move at the point that is becomes it center and all the other points of the center are on common distance. So the figure that are rotated from any point will change its facing side means if we want to change the face of the figure then we use the rotation method of transformation .

In upcoming posts we will discuss about Special right triangles and Methods of data representation. Visit our website for information on CBSE home science syllabus

Coordinate geometry

In mathematics the coordinate geometry have a great importance. So we are going to discuss here the coordinate geometry. To understand the coordinate geometry you should be familiar with some related terms of coordinate geometry. The grid is a combination of horizontal (Find Horizontal Asymptote) and vertical lines that makes the squares. On the grid squares two points are there called axis, the x- axis and y- axis, the x axis represents the horizontal line and y axis represents the vertical lines. Both the points are intersecting on some point that is O point and called the origin. In which we determine the distance between the two points or the simple definition of the coordinate geometry is that, it is used mainly to address the point that are on the plane by using numbers of pairs that should be in ordered form.
Coordinate geometry formulas are used to define some coordinate geometry topics.
To determine the distance between the two pints we use distance formula that is
Distance d = √ [(x_{2 –} x₁)² + (y_{2 –} y₁)² ],
To find the angle between the two lines, we have formula θ tan – ¹m, where ‘m’ is slope, an angle may be anyone like right, acute, obtuse etc.
The slope formula is m = (y₂ – y₁) / (x₂ – x₁), we can also find the slope for perpendicular and parallel lines.
The midpoint formula (x, y) = (x₁ + x₂ / 2, y₁ + y₂ / 2),
We can also find the area and perimeter of a polygon in the field of geometry that is defined by the points. We can also transform the shapes in the coordinate geometry. It is used mostly in real life constructions.


In upcoming posts we will discuss about Rotations and Statistical experiments. Visit our website for information on CBSE previous years 11 physics

reflections

In this math helper blog we are going to discuss about the different types of laplace transformations in which one is reflections but first of all we have to know about the meaning of transformation that, it is defined as any change in the shape, size or orientation of the thing and According to the definition of transformation it is the way of changing the shape or appearance of the figure is known as the transformation of the figure. One of the methods of transformation is reflections that are defined as follows.
Reflection: It is also one of the methods of transformation in which mirror image of the things will created. In the reflection always side of the thing will changed means it will appear on its opposite side and also the shape of that thing will not change. One more thing should be noted that if we define reflection mirror image of the particular thing will make on the same line as original thing and also on the same line on which actual thing is situated. A mirror line either horizontal or vertical it will not change the shape of the thing .So according to the reflection definition when applied a method that provide the mirror image of the figure is known as the reflection that is the one of method of transformation . For more details for reflection click here
There are many example of reflection that are as echo of noise is one of the common example of daily life .Reflection is used in many technologies as SONAR, radar etc. All the waves as radio waves or electromagnetic waves are work on the method of reflection .So the reflection will give the many of the important application that is mostly used in our life.

In upcoming posts we will discuss about Coordinate geometry and Correlation and causation. Visit our website for information on CBSE political science board paper

trigonometric ratios

Hello students, in this blog we are going to discuss the trigonometric ratios. The trigonometric ratios of angles are used to find the angles and sides for a triangle when we have some of angles and sides are given. This task is the main task of the trigonometric ratios. This problem is solved by using some ratios of the sides of a triangle with respects to its acute angles. These ratios of acute angles (What is an Acute Angle) are called the trigonometric ratios of angles.
Note: - The trigonometric ratios are same for the same length.
We have six trigonometric ratios; let’s take a look on them.
With reference to an angle ‘A’ in a right angled, ΔABC, right angled at ‘c’.
‘a’ is the opposite side (Perpendicular)
‘b’ is the adjacent side (Base)
‘c’ is the hypotenuse.
The ratios of sides a / c, b / c, a / b, b / a, c / b, c / a have the following names, they are : -
a / c is called the sine of ‘A’, written as sin A.
b / c is called the co-sine of ‘A’, written as cos A.
a / b is called the tangent of ‘A’, written as tan A.
b / a is called the co-tangent of ‘A’, written as cot A.
c / b is called the secant of ‘A’, written as sec A.
c / a is called the co-secant of ‘A’, written as cosec A.
Thus, we have six trigonometrically ratios which must be memorized
sin A = perpendicular / hypotenuse,
cos A = base / hypotenuse,
tan A = perpendicular / base,
cosec A = hypotenuse / perpendicular,
sec A = hypotenuse / base,
cot A = base / perpendicular,


In upcoming posts we will discuss about reflections and Permutations and combinations. Visit our website for information on business studies class 12 CBSE syllabus

circles

Circle is the locus of a point which moves in a plane in such a way that its distance from a given fixed point is always constant. This fixed point is called center of circle. Practice on circles by Finding the Area of a Circle
 Circles geometry can be explained as shown below,
A line segment joining the end point on the circle is called its radius. The plural of radius is radii.
A chord of a circle is a line segment joining any two points on the circle.
A diameter is a chord of circle passing through the center of circle.
The perimeter of a circle is called its circumference.
A line which intersects a circle in two distinct points is called a secant of the circle.
Theorem used in circles geometry is shown below,
Theorem: Equal chords of a circle subtend equal angle.
Given a circle C (O, r) in which chord AB = chord CD. To Prove that AOB = COD, we have OA = OC.
Sector of circle is enclosed by an arc of a circle and the two bounding radii. The diameter of circle divides the circle into two equal arcs; each of these two arcs is called a semicircle.
Circles which have the same center and different radii are called concentric circles (for more on circle click this). The different places of the circle are discussed below:
1. inside the circle
2. on the circle
3. outside the circle
Some other properties of circles,
The angle in the semicircle is a right angle which is 90°.
The arc of the circle subtends a rights angle at any point on the circle.
Angles in the same segment of a circle are equal.
Above discussion is helpful for Grade XII students to understand the concept of Circles.


In upcoming posts we will discuss about trigonometric ratios and Math Blog on Grade XI . Visit our website for information on CBSE syllabus for class xi english core

Basic constructions

Hello students, in this session we are going to learn about the Basic Constructions in geometry. In mathematics the geometry help plays a very important role because in this we teaches how to make shapes, figures, lines and angles. A construction in geometry is the skill by which we can able to draw the many types of figures. All constructions will be completed only by the use of ruler and compass as instruments. In every construction we are expected to write down the essential steps of constructions.
Note: - Never draw freehand when doing constructions. (Also refer algebra equation solver to improve your skills)
The list of the basic constructions in geometry is: -
-Copy a line segment.
-Copy an angle.
-Bisect a line segment.
-bisect an angle.
-Construct parallel lines, perpendicular lines and many more.
Mainly we include line, circle and triangle constructions in the geometry constructions.
Line construction is the basic construction for the all figures in geometry. By joining the line we can construct the many figures. And for making the straight line we use ruler and pencil.
Circle construction can be form by using the two ways that are diameter and radius. For making the circle we should have either diameter or radius of the circle. The following steps for constructing the circle: -
Step 1: - Set the compass with the ruler according to required radius.
Step 2: - Then set up the compass' end point on the paper.
Step 3: - Now put the end of the pencil on paper.
Step 4: - Revolve the end of pencil in any direction until we meet the starting point.
Triangle is constructed using ruler, compass and protractor.
Grade XII students can learn basic geometry constructions using the above information.

In upcoming posts we will discuss about circles and Probability and Statistics. Visit our website for information on CBSE 11 physics book

Triangle congruence relationships

Hello students, in this answer math problems for free session we are going to read about the triangle congruence relationships. But before discussing it we will discuss Congruent Triangles, it means âˆ†ABC is said to be congruent to ∆DEF only when one of them can be made to superpose on the other (and vice-versa ) so as to cover it exactly. And, we write ∆ABC ≅ âˆ†DEF. The congruence relation for the triangles is:-
-Every triangle is congruent to itself that is âˆ†ABC ≅ âˆ†ABC.
-If ∆ABC ≅ âˆ†DEF then ∆DEF ≅ âˆ†ABC.
-If ∆ABC ≅ âˆ†DEF then ∆DEF ≅ âˆ†PQR, then ∆ABC ≅ âˆ†PQR.
There is some criteria by which we can also show the triangle congruence relationships, the criteria are: -
SAS (side - angle – side): - If two triangles have two sides and the included angle of the one equal to the corresponding sides and the included angle of the other, then the triangle are congruent.
ASA (angle - side – angle): - If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle, then the two triangles are congruent.
AAS (angle – angle – side): -If two angles and any side of a triangle are equal to the corresponding angles and side of another triangle than the two triangles are congruent.
SSS (side – side – side): - if the three sides of one triangle are equal to the corresponding three sides of another triangle than the two triangles are congruent.
RHS (Right – angle – Hypotenuse – Side): - Two right angled triangles are congruent if one side and the Hypotenuse of the one are respectively equal to the corresponding side and the Hypotenuse of the other.
Above discussion helps Grade XII students to understand Triangle congruence relationships.

In upcoming posts we will discuss about Basic constructions and Types of events. Visit our website for information on CBSE 10th science syllabus

triangle inequality theorem

Hello students, in this session we are going to discuss the triangle inequality theorem. This theorem defines that the sum of any two sides will always be greater than the third side or we can say that the only one side is shorter than the other two sides; meaning is same in both the scenario. The theorem is - If we have x, y and z sides for the any triangle then,
x + y > z,
y + z > x,
x + z > y,
It is to be noted that if any one side of triangle is greater than the other two sides then we cannot construct the triangle. Also the triangle inequality theorem says same. for more information visit here
We can see this by an example:
Is a triangle with the sides 6cm, 7cm and 8cm possible?
Solution: Sum of 2 sides is always greater than the third side.
Then according to theorem
x + y > z,
y + z > x,
x + z > y,
6 + 7 > 8,
7 + 8 > 6,
8 + 6 > 7,
We can see that in above example the sum of 2 sides is greater than the third side. So, the triangle is possible.
Is a triangle with 3 cm, 4 cm and 13 cm possible?
Solution:
3 + 13 > 4
4 + 13 > 3
3 + 4 >12
In above example the last inequality is false, so, the triangle is not possible
Note:
 i) If in two triangles the two sides are congruent, then the triangle that have larger third side will keep larger included angle.
ii) If in two triangles the two sides are congruent, then the triangle that have larger included angle will keep a larger side.
Central Board of Secondary Education Grade XII students can practice by reading this discussion.

In upcoming posts we will discuss about Triangle congruence relationships and Permutations and combinations. Visit our website for information on Circumference Formula of a Circle

parallel lines cut by a transversal

Hi Friends! In this online tutors homework help session we will talk about parallel lines cut by a transversal. Let us consider two lines which are at equidistance from each other at any point of observation. It means that when we draw a perpendicular at any point from one line to another, we observe that all perpendiculars are of equal length. In this unit we will discuss about the topic parallel lines cut by a transversal. If two parallel lines are intersected by a line which intersect both the lines, then it is called a transversal. To understand parallel lines cut by a transversal definition, we say that a transversal is the line which meets two parallel lines at some point. So a transversal has a point of intersection on both the parallel lines.
Let us consider two parallel lines say ‘l’ and ‘m’. Let ‘n’ be the transversal drawn on the two parallel lines. Now the following   conditions are satisfied:
1.      Since we have l|| m, then the corresponding angles formed by the transversal are equal. These types of angles are four in pairs.
2.      The pair of interior opposite angles is supplementary. These angles are 2 in pairs.
3.      The pair of interior alternate angles so formed is also equal. These angles are 2 in pair.
4.      Also exterior alternate angles are equal. They are also 2 in numbers.
Now keeping these qualities in mind, if one of the angles among all the angles is known, we can find rest of the angles so formed. To find these angles we may use the property of vertical opposite angles are equal, corresponding angles of the two parallel lines, cut by a transversal are equal, Linear pair and the property of alternate angles.
Besides this we come across the problems where we are given some of the measures of the angles (also see Complementary Angles Definition) and we need to find if the two lines which are cut by the transversal are equal or not. This discussion will help students of grade XII to understand the concept of parallel lines cut by a transversal.

In upcoming posts we will discuss about triangle inequality theorem and Measures of central tendency. Visit our website for information on biology syllabus for class 10 ICSE

Angles of triangles and polygons

We say that a polygon is a closed figure with three or more sides. If we talk about What is the Area of a Triangle, we say that a triangle is a closed figure or we call it a polygon with three sides. Here we observe that a triangle has three sides and so it has three angles. Also we must remember that the sum of Angles of triangles is 180 degrees. Thus if we have all the three angles equal, then each angle of the triangle = 180/3 = 60 degrees. Also try area of equilateral triangle calculator to sharpen your skills.
 Now we will look at other polygons. Let us take a three sided figure, say a quadrilateral. We know that a quadrilateral is a four sided figure and it has 4 angles. We also must remember that a quadrilateral is formed by joining 2 triangles. Now if the sum of angles of a triangle is 180 degrees so we say that the sum of angles of a quadrilateral is 180 + 180 = 360 degrees. Thus all the quadrilaterals have an angle sum of 360 degrees.
 Now a polygon can be any figure with 3 or more line segments and when we need to find the Angles of polygons, we must always remember the following formulas:
If the regular polygon is of sides ‘n’, then we sum of have:
1)     Each exterior angle = 360⁰ / n.
2)     Sum of all exterior angles of any polygon = 360⁰.
3)     Also we have each interior angle = 180⁰ - (each exterior angle).
In case of complex polygon of n sides, we have the sum of all exterior angles = 4 right angles = 4 * 90 degrees = 360 degrees.
Also sum of all interior angles = (2*n -4) right angles.
These formulas will help us to find the angles of all the polygons.

In upcoming posts we will discuss about parallel lines cut by a transversal and Conditional probability. Visit our website for information on syllabus of economics for ICSE class 12

Properties of quadrilaterals

In this unit we are going to learn about the Properties of quadrilaterals. The quadrilaterals are the four sided closed figures which are formed by joining the four line segments. All squares, rectangles, parallelograms, trapezium, kite and even all irregular four side figures are called quadrilaterals. A regular quadrilateral is called a square. Here we are going to learn about Define quadrilaterals properties.
We first look at a square: It is a four sided figure with all the sides equal. All the angles of the square are 90 degrees. So we say that it has opposite sides parallel and equal.
Rectangle: A rectangle is a quadrilateral with its opposite sides equal and parallel. Here we have to remember that all the angles are 90 degrees as in square, but all sides are not equal. So we can say it is a square is a special rectangle with its length and breadth as equal.
Parallelogram: A quadrilateral is a four sided figure with its opposite sides parallel and equal. All squares and rectangles are parallelogram, but it is not necessary for all the parallelograms to have its angles as 90 degree, so we conclude that all parallelograms are not necessary a square or the rectangles.
In case of trapezium, we have one pair of opposite sides as parallel, but the pair of parallel lines is not equal. So we come to the conclusion that another pair of opposite lines formed in the trapezium is neither parallel nor equal.
 If we look at a kite, it also has 4 sides so it is called a quadrilateral (for more). In kite we have a pair of adjacent sides equal instead of the pair of opposite sides.
Rhombus is a figure with all four sides equal. It is a tilted form of the square. It has the pair of opposite angles equal but not equal to 90 degrees. In rhombus, we have the diagonals are perpendicular bisector. This discussion will help students of grade XII to understand the Properties of quadrilaterals.

In upcoming posts we will discuss about Angles of triangles and polygons and Mean. Visit our website for information ICSE board syllabus for class 12 math

Congruence

The topic congruence occurs in geometry and we say that two figures are said to be congruent triangles if they are same in shape and size, which means that if either object is repositioned, it should coincide precisely to the other figure.
For the students of grade XII here is an example: Here are two triangles which are congruent as their corresponding sides are same in length and Corresponding Angles Definition are same in measure.

If congruence occurs in the above two triangles, then mathematically we can write the relationship as:
∆ ABC ≅ ∆ DEF
The symbol ‘≅’ denotes “is congruent to”.
When we study the properties of congruence, then we realize that the order of the points is important. That means the triangles will coincide if A is placed on D, B on E and C and F.
Whereas, the topic congruence does not go right if we say  âˆ† ABC ≅ ∆ EFD.
As all the grade XII students can observe that the topic congruence fits in the above example as:
The corresponding sides are equal, i.e., AB = DE, BC = EF and CA = FD and corresponding angles are also equal (∠ A = ∠D , ∠ B = ∠ E, ∠ C = ∠ F ).
One can note that formally, congruence for the set of two points goes like: they are congruent if and only if when one point can be repositioned or transformed into the other point by an isometry, i.e., by rotations, translations and reflections.
To all grade XII students, now we are moving ahead in the topic congruence to the principles of congruent triangles:
1.       The SSS principle: As two triangles are equal if all the corresponding sides are equal.
2.       The SAS principle:  Two triangles are congruent if two pairs of corresponding sides are equal and the angles included between them are also equal.
3.       The ASA principle: Two triangles are congruent if two pairs of corresponding angles are same in measure with one pair of equal corresponding sides.
4.       The RHS (Right Angle Hypotenuse Side) principle:  Two right angled triangles are congruent if the hypotenuses of both the triangles are equal with a pair of equal corresponding sides.


In upcoming posts we will discuss about Properties of quadrilaterals and Mean. Visit our website for information on ICSE syllabus for class 3 maths

Learn Pythagorean Theorem

Hello friends today we are going to learn about Pythagorean Theorem also known as Pythagoras Theorem

Pythagorean Theorem Examples: This theorm was derived by a greek mathatician Pythagoras .It is related to all sides of a Triangle and can be applied to those triangle who is having one 90 degree angle.• Arms (a and b): the sides of the triangle adjacent to the right angle. They should not be of same length to apply Pythagorean theorem
• Hypotenuse (c): the side of the triangle opposite the right angle
Theorm :
Equation :Equation :a²   +   b²    = c²

The sum of the squares of the two sides is equal to the square of the hypotenuse


 Lets take an acute angle triangle abc here angle a and b are acute angles a and b angles are acute angles less than 90 degree and one angle i.e c is of 90 degree to give a sum of 180 degrees.In this angle there is no obtuse angle (angle greater than 90 degree)to apply the theorm
This theorm is very useful for finding the any side of a right angle triangle . If the length of any side is not known it can be calculated by using
Where a and b represents length of other two sides and c represents length of the hypotenuse ,the longest side
Pythagorean theorem help with more Examples. (visit for detail)
Example 1: Prove a triangle with sides “2, 3 , 4” is having a 90 degree angle in it
Solution = 2*2+3*3 =4*4

Example2 :Find the side of the Triangle(12,5  , c) ?
In this question values and info we are already having are two side length and one angle is of 90 degree.So to calculate the length of the third side we can apply the Pythagorean theory
A² +b ²=c²
12*12+5*5=c²
144+25=c²
169=c²
13=c

This theorm will be helpful to everyone till grade XII

In upcoming posts we will discuss about Congruence and Applications of Probability and Statistics. Visit our website for information on Karnataka state board books