Geometry Triangle Congruence

Triangle is a polygon with three sides.Congruent triangles are special types of similar triangle which have both shape and size equal. If one triangle is placed on the other triangle it is fitted correctly on the first triangle.Two triangles are congruent if all parts of both the triangles are the same.

A triangle has 6 parts

3 sides
3 angles
So when two triangles are congruent all the 6 parts are congruent .

Introduction to geometry triangle Congruence:
In  two dimensional Plane Geometry, two triangles are congruent if all the corresponding parts of those triangles are similar.  The corresponding sides of two triangles are equal in measure and their corresponding angles are equal in degrees, we can say the two triangles are congruent.If triangle PQR is congruent to triangle XYZ, the relationship between these two triangles can be written as:?PQR `~=` ?XYZ

Congruence of Triangles in Geometry:
Postulates based on congruence of triangles:

Angle Angle Angle (AAA)

Side Side Side (SSS)

Side Angle Side (SAS)

Angle-Angle-Side(AAS)

Right-angle-Hypotenuse-Side(RHS)

Angle Angle Angle (AAA):

When three angles of the two triangles are equal, we can say that the two triangles are similar triangles.That is the corresponding angles have same measures.

Side Side Side (SSS):

When all three sides of both the triangles are equal, we can say that the triangles are similar triangles.

Side Angle Side (SAS):

When two sides in one triangle are equal to corresponding sides of the other triangle, and the included angles are equal, we can say that both are congruent triangle.

Angle-Angle-Side (AAS):

When two pairs of angles of two triangles are equal in measurement, and pair of corresponding unincluded sides are equal in length, then the triangles are called as congruent triangle.

Right-angle-Hypotenuse-Side (RHS):

When two right-angled triangles with their hypotenuses equal in length, and the shorter sides of two right triangles are equal in length, then the triangles are called as congruent triangle.

Properties of Congruence Triangles in Geometry:

If two triangles are congruent triangles in geometry, then each side or angle of the triangle is congruent to the corresponding part in the other triangle. Once if we proved two triangles are congruent, we can find the angles or sides of one of them from the other triangle.

To remember this we use the acronym CPCTC, which is expressed as “Corresponding Parts of Congruent Triangles are Congruent". In addition with sides and angles of two triangle, all other properties such as area, perimeter, location of centers, circles etc. of the triangle are the same.

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Example Based on Congruency of Triangles in Geometry:
Ex : 1From the figure state that the triangles are congruent or not . If congruent by which postulate

Sol:From figure
QR = 4cm = TV

PR = 5cm = SV

Two sides and the included angle are congruent

So by SAS postulate the triangles are congruent.

Solving Equations by Factoring

Introduction to solving equations by factoring:
What is an Equation ?

An equation is a mathematical representation of two expressions which are equal to each other. Every equation has two parts one is the

Left Hand Side and other is the Right Hand Side. The left hand side is connected to the right hand side by an equal sign.

Properties of an Equation

1) Any quantity can be added to both the sides of an equation.

2) Any quantity can be subtracted to both the sides of an equation.

3) Any quantity can be multiplied to both the sides of an equation.

4) Any non zero quantity can divide both the sides of an equation.


Solving Equations by Factoring - Different Methods

Equations can be solved in many ways.

1) By means of factoring.

2) By taking the roots.

3) By completing the square in case of quadratic equations.

4) By using the quadratic formula.

5) By using the method of graphing.

In this article we will discuss solving equations by means of factoring.

In this context lets first have the idea of ' Zero Factor Principle ' .

What is Zero factor principle ?

Zero factor principle states that product of two expressions is zero , if and only if either of the expression is zero.

Explanation : Let A and B be two expressions.

Product of A and B is AB.

AB = 0 , the relation holds true if and only if either A = 0  or B = 0 .

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Solving Equations by Factoring-examples

Now lets see some examples.

d  a = 5 / 3.

Examples :

1 )    x ^2   -  5x   -  6  = 0

Or,    x^2 - 6x + x - 6  = 0    [ Always break the middle term , such that even after breaking there is no change in the expression ]

Or,    x ( x - 6 ) + 1 ( x - 6 ) = 0

Or,    ( x - 6 ) ( x + 1) = 0

Now from Zero factor principle we can conclude that either ( x - 6 ) = 0   or  ( x + 1 ) = 0

When ( x - 6 ) = 0                                           When ( x + 1) = 0

Or,       x   =   6                                                           Or, x = - 1

Therefore the equation has two roots x = 6  and x = - 1

Explanation

Procedure to break the middle term.

First we will find the product of the coefficient of x^2 and the constant term in the equation.

In this case it is 1 * ( - 6 ) = -6

Now we will find two numbers that multiply to give -6 ( product of the coefficient of x^2 and the constant term )

and add to give -5 ( coefficient of x ).

The probable numbers are

a)  2 and -3 :     2 * ( - 3) = -6   but 2 + ( - 3 ) = -1 so it will not be accepted.

b)  - 2 and 3 :    ( - 2 ) *  3 = -6   but ( - 2 ) +  3  = 1 so it will not be accepted.

c)  6 and -1 :     6 * ( - 1) = -6   but 6 + ( - 1 ) = 5 so it will not be accepted.

d)  -6 and 1 :     1 * ( - 6 ) = -6   but 1 + ( - 6 ) = - 5 so it will  be accepted.

So that is why the middle term is represented as 5x  = - 6x + x .

2)     x^2   -  7x   +  12  =  0

Or,   x^2  -  4x   -  3x   +  12 = 0       [ Always break the middle term , such that even after breaking there is no change in the expression ]

Or ,   x ( x - 4) - 3 ( x  - 4 ) = 0

Or,    ( x - 4 ) ( x - 3 ) =  0

Now from Zero factor principle we can conclude that either ( x - 4 ) = 0    or   ( x - 3 ) =  0

When   ( x - 4 ) = 0                                                     When   ( x - 3 ) = 0         

Or,        x  =  4                                                                        Or,  x  =  3

Therefore the equation has two roots x = 4  and x = 3

Explanation

Procedure to break the middle term.

First we will find the product of the coefficient of x^2 and the constant term in the equation.

In this case it is 1 *  12  =  12

Now we will find two numbers that multiply to give  12( product of the coefficient of x^2 and the constant term )

and add to give -7 ( coefficient of x ).

The numbers are   - 4   and  - 3 :

( - 4 ) * ( - 3 ) = 12   and   ( - 4 ) + ( - 3 )   =   - 7   so it will be accepted.

So that is why the middle term is represented as   - 7 x  =   - 4x   -  3 x .

3)     6a^4  -  13a^3  + 5a^2  =  0

Or,    a^2 ( 6a^2 - 13a + 5 ) = 0

Or,    a^2 ( 6a^2 - 10a  -  3a  +  5 ) = 0

Or,    a^2 { 2a ( 3a - 5 ) - 1 ( 3a - 5 ) } = 0

Or,    a^2 ( 2a - 1 ) ( 3a - 5 )= 0

Now from Zero factor principle we can conclude that  a^2 = 0   or  ( 2a - 1 ) = 0    or   ( 3a - 5 ) =  0

When ( 2a - 1 ) = 0                                                   When ( 3a - 5 ) = 0                                  When a^2 = 0

Or,        2a = 1                                                                Or,       3a  =  5                                        Or, a = 0 

Or,          a =  1 / 2                                                           Or,        a = 5 / 3

Therefore the equation has  roots as a = 0  ,  a = 1 / 2  an

Decomposing and Composing Numbers

Introduction :

Decomposing and composing numbers is the very old concept in number system and mathematics. It means to take numbers apart into their tens and ones form. The smaller problem should be simpler than larger problem, so by using decomposing, large problems into simple problems. Larger problems can be tackled with ‘divide and conquer”. So we are using decomposing and composing numbers.


Discussion on Decomposing and Composing Numbers

Decomposition:

Decompose the problem so that:

Each sub problem is at the same level of detail,
Each sub problem can be solved independently,
The solutions to the sub problems can be combined to solve the original problem.
Composition:

Each sub problem is at the different level,
Each sub problem can be solved dependently not to be easy one,
The solutions to the sub problems cannot be combining to solve the actual problem.


Advantages of decomposing and composing numbers:

Different people can workout on different sub problems,
Parallelization may be possible,
Maintenance is easier.


Disadvantages of Decomposition and composition:

The solutions to the sub problems might not combine to solve the original problem,
Poorly understood the problems are hard to decompose.

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Some Example Problems on Decomposing and Composing Numbers

To begin the decomposing numbers you must take a 2- digit number apart, like this.

45 = 40+5

So, 45 is the same as saying 40+5, because the 4 in 45 stands for 40,

And the 5 in 45.

45 stand for 5,

Example 1: 53 + 25 =?

First add the tens ….. 50 + 20 = 70

Add the ones …            3 + 5   = 8

Then add the sums! And get the answer as,

70 + 8 = 78

Example 2: 154 + 32 =?

First add the hundreds, and then add tens and finally add ones….

100 + 50 + 30 + 4 + 2 totally add all, the sum is,

100 + 50 + 30 + 4+ 2 = 186, so get the answer is 186.

Example3: Choose the correct expression that will result in 818.

a)      200 + 200 + 200 + 10 +8

b)      250 + 250 + 200 +10 + 8

c)      200 + 200 + 200 + 200 + 10 + 8

d)      200 + 200 + 200 + 18

Solution:

Answer is c) 200 + 200 + 200 + 200 + 10 + 8

Steps to derive:

Simplify given answer chose one by one

200+200+200+10+18 = 618, and

250+250+200+10+8 = 718, and

200+200+200+200+10+8 = 818, and

200+200+200+18 = 618.

Hence the right option (choice) is “c”

So the answer is 200+200+200+200+10+8 represents = 818.