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

Intersection of a plane with 3-d figures

Intersection of a plane with 3-d figures for Grade XII of Karnataka state board syllabus:
Some interesting points are there related to Intersection of a plane.
• If there are three collinear points then there is only one plane that consists of all three points.
• If two different planes have a common point then definitely the intersection of those planes would be a line.
• Two planes would be parallel if they have no common points or they are identical.
• A plane section of a 3D figure would be an intersection of that figure with a plane.
• If a line l is perpendicular to two different lines m and n at their point of intersection then  that line(l) would be perpendicular to the plane that contains those lines(m and n).
In 3D coordinate system a point is defined with the three coordinates which indicates its position from the all three axes. There are three coordinate systems
• Cartesian coordinate system
• Cylindrical coordinate system
• Spherical coordinate system
You can also improve your math skills by reading Parallel and Perpendicular Planes. In Cartesian coordinate system appoint P is denoted as P(x, y, z). The three axes in all three coordinate systems are perpendicular to each other.
In cylindrical coordinate system a point P is denoted by the points P(r, ᶲ, z). for ex ample we can choose a ray originating at the intersection at the plane and the axis( the origin). Here r is the radius of the cylinder, ᶲ is the angle between x axis and the cylinder’s radius and z denotes the z coordinate.
For transforming cylindrical coordinates to Cartesian coordinates or vice versa transform equations are used.
X = r cos θ; y = r sin θ and z = z. Visit this for more information.
Spherical coordinates are denoted as P(r, θ, ᶲ). The same for transition of one form to other transition equations are needed.

In upcoming posts we will discuss about Learn Pythagorean Theorem and quartiles. Visit our website for information on Binomial Probability

Geometric proofs in Grade XII

Hello students. I am going to tell you about Geometric proofs in Grade XII of Karnataka board syllabus for various shapes like triangles, circles and many more. Geometry considers all the measurements of a figure like shape, size, and associate position of figures. Geometric proofs are the step by step process from postulates of a proof to outcomes. There are two most approaches that works for proofs of geometry. They are direct proof and indirect proof. Subsidiary lines are draft along with the proofs. Also improve your knowledge in Introduction to geometry.
For geometric proofs you have to follow some terms that are used frequently. They are:
-Auxiliary lines:-Created to provide support to assure statement.
-Contradiction:- This case arises when the negation of a true expressions is also true.
-Direct proof:- In this we can directly take conclusion from the antecedent conclusions.
-Indirect proof:- When it follows contradiction.
-Paragraph proof:-In this proof we can written all steps into single expression in a paragraph form.
-Two columns proof:-In this type of proof we construct a two column table the first column consist all conclusion and second column consist the reasons for each conclusion.
-Geometric proof:- It is a step by step procedure that used all of the terms.
Note:- paragraph and two column proof are not same although they have same in content.
I am giving you a geometric proofs example so that you can understand it in a better way:-
We have a figure:


in given figure Section PR intersect SQ
Section QS intersect RP.
So you need to prove triangle PQZ and RSZ are congruent.

Statements	Reasons
1.Section PR intersect SQ.	1.Given
2.Sections PZ and RZ are congruent.	2.When a section is intersected then two resulting section are congruent.
3. Section QS intersect RP.	3.Given
4. Sections QZ and SZ are congruent.	4. When a section is intersected then two resulting section are congruent
5.Angles PZQ and RZS are congruent.	5.Verticals angles are congruent.
6.Triangles PQZ and RSZ are congruent	6.SAS postulates.

In upcoming posts we will discuss about Intersection of a plane with 3-d figures and dispersion. Visit our website for information on Area of a Circle Formula

Conditional statements

Hi Friends! In this session of solve math problems we will discuss about conditional statements. Conditional statements are the statements that check the true and false value for every given statements. In Conditional statements some statements shows that they are universally true (like:-sun rises from east) and some are false. In free online math tutoring sessions Conditional statements are generally same in every grade either grade XII or others. These statements are used in computer science in programming languages.
Conditional statements are:-
-If then else conditional statements are used to execute if block when this is true if it is not true than it goes to the else block and execute it syntax for it:-
if(condition)
statement1;
else if(condition2)
statement2;
-Else if is the nested if else block that have multiple else blocks.
if(condition1)
statement1;
else if(condition2)
statement2;
else if(condition3)
statement3;
else
statement-n;
-Switch statements are also used to check multi way conditional statements.
switch(expression)
case choice1:
statement1;
case choice2:
statement2;
case choice-n:
statement-n;
-Conditional operator (?) is also used to check true or false value for given statements syntax for this is:-
Condition ? Expression 1 for condition true : Expression 2 for condition false
Many operators are used to check statements like :-
logical operator:- AND, OR, NOT, XOR.
Comparison operator:- =, <=, >=, !=.
Conditional statements examples can be understand by converse, inverse and contra positive expressions.
If the deer will run then tiger will hunt the deer---conditional.
If the deer won't then the tiger won't hunt the deer---inverse.
If the tiger will hunt the deer then the deer will run---converse.
It the tiger won't hunt the deer then the deer won't run---contra positive.
If p then q conditional
if q then p converse
if p will be negation than negation q inverse
if negation q then negation p contra positive
Logically equivalent will be conditional expression and its contra positive.

In upcoming posts we will discuss about Geometric proofs in Grade XII and Correlation coefficient. Visit our website for information on school education Karnataka

Formal/informal proofs

Hi Friends! In this session of solve math problems for free, we will discuss about formal and informal proofs. Proofs are the statements that are said to be true in every case. When a statement is proved to be true then it is considered a theorem. Proofs are widely used in various fields of mathematics and they can be categorized in two types: Formal and informal proofs. Formal proofs follow step by step procedures and rules for proving statements, all work on formal proofs is done systematically, they cannot be written in natural language although they have own formal language that contains some rules, syntaxes, and predefined approaches that are written in English language.You can also refer free online math help
In formal proofs we find the solutions for every step with the reasons. In informal proofs we also have to find solutions but there is no necessity to define reasons for that. In the informal proofs our focus is only on solutions not on solution procedures and in formal proofs our focus should be based on solutions as well as solution procedures.
Conditional statements are the statements that are mainly based on true and false conditions and used in various programming languages they have some procedures that are used to obtain the sentences value like:-
-If then else, in this we have two statements one is in if block and second is in else block. When If condition is true than if block will be executed and when else condition is true than else block will be executed.
-Else if, it is nested if else block that is used for various conditions. Click here for more on proof theory.
Switch statement and ternary operator are also used for conditional statements. To verify conditional statements lots of operators are used like logical operator like and, not, or and comparison operator like =, <=, >= and many more. All operators have their different-2 syntax mechanism to solve various conditional statements.


In upcoming posts we will discuss about Conditional statements and Probability problems with finite sample spaces. Visit our website for information on Karnataka state board

Euclidean/non-Euclidean geometries in Grade XII

Hi Friends! In this online geometry help session we will discuss geometry and it's types. Geometry is basically classified in two types namely: Euclidean and Non-Euclidean geometries. Euclidean Geometry refers to a high school geometry that deals with straight lines, planes and points, whereas non Euclidian geometry deals with non straight lines which are nothing but curved lines. Euclidean geometries and Non Euclidean geometries are the opposite forms of mathematical geometrical methods in geometry. Euclidean/non-Euclidean geometries in grade XII consists of the understanding and application of many forms of geometrical constructions. The term Euclidean geometry is named after its invention by an ancient Greek mathematician-Euclid (300 BC).Euclidean geometry mainly consists of any two straight lines which follow a path parallel to each other.(parallel means lines which face each other and flow constantly and  remain at an equal distance from each other even if they are extended to any point, parallel lines never intersect each other.)Non Euclidean geometry is of many types which consist of curved lines and the lines may flow in a perpendicular direction with respect to each other. Examples for non Euclidean geometrical diagrams are elliptical forms, hyperbolas etc. The latest or the modern forms of Euclidean's theory are of four types which are most commonly used and known. They are the Pythagoras theorem, Thales theorem, Bridge of asses theorem and sum of angles theorem which states that the sum of all the angles of any triangle is always equal to 180 Degrees. Simple examples of Euclidean geometry are squares, rectangles, quadrilaterals. A parallelogram is another example that has two sides parallel to each other and are measured and drawn using scales, where as Non Euclidean Geometrical examples include spherical shapes, elliptical shapes, which are in the form of a curve.
HYPERBOLA is a practical Example for Non euclidean geometrical (for more see this)form which consists of the line that touches the x axis and passes through the y axis.It meets all the criteria of a non euclidean geometrtical form like being in curved way,or non striaght form that cannot be drawn using a scale.

Here  we have a wonderful example for euclidean geometry, which consists of parallel lines in the rectangle (also see What is the Area of a Rectangle)given.in the next topic we are going to discuss formal and informal proofs.
In the next topic we are going to discuss Formal/informal proofs

In upcoming posts we will discuss about Formal/informal proofs and variance of discrete random variable. Visit our website for information on Maharashtra state board books

Geometry and Measurement

Geometry and measurements is the most important part of the mathematics. Geometry word is made from Geo and metria(Geo means earth and metria means for measurement) in this we are define the whole measurement of the earth. In this topic we are discussing the shapes, angles and turns, measuring length, perimeter, area, cubes, surface area and volume.

In analytic geometry problems we define a shape and its measurement. In the above shape five variables are use and all variables are connected to each other and make a triangle. In this diagram the three angles are of same length and some lines are symmetric, and one line is connected to the point A and point E, and makes a triangle.
In Geometry, measurements are used to describe all the circles and curves. In geometry we calculate the areas and circumferences of figures such as circles, rectangles, cubes, cuboids, etc.
The Geometry and measurement are mainly used to find the location of area on earth, when we go to an unknown area, we can see the map and find the proper location. In this we find the angle of any value.
When we draw a chart on a graph paper on any equation, we are actually using concepts of geometry. In grade XII students should know how to find total surface areas, curved surface areas, circumferences etc of figures like triangles, rectangles, pentagons, circles and other polynomials.

In upcoming posts we will discuss about Euclidean/non-Euclidean geometries in Grade XII and Central limit theorem. Visit our website for information on Maharashtra board syllabus class 11