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Concepts of Geometry

Triangle

Definition : A triangle is a polygon with three sides.

                    It also has three vertices and three angles.

Types of triangle:

1. Equilateral Triangle

    An equilateral triangle is a triangle in which all three sides are the same length.

2. Isosceles Triangle:

    An isosceles triangle is a triangle in which two sides are the same length and the third side is a different length.

3. Scalene Triangle:

    A scalene triangle is a triangle in which all three sides are different lengths.

4. Acute Triangle

    An acute triangle is a triangle in which all three of the angles are acute angles.

 

5. Obtuse Triangle

    An obtuse triangle is a triangle in which one of the angles is a obtuse angle.

6. Right Triangle

    A right triangle is a triangle in which one of the angles is a right angle.

• Pythagoras theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

Properties

• The sum of the interior angles of a triangle is 180 degrees.
• The sum of the exterior angles of a triangle is 360 degrees.
•  The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior     angles of the triangle.
•  Each interior opposite angle always less than that of exterior angle.
•   Angles opposite to two equal sides of a triangle are equal.
•  Sides opposite to two equal angles of a triangle are equal.
•  In an isosceles triangle altitude bisect the base.
•  If two sides of a triangle are unequal, then the longer side has greater angle opposite it.
• The greater angle of a triangle has the longer side opposite to it.
• The sum of any two side of a triangle is greater than the third side.
•  In a right angle triangle, the hypotenuse is the longest side.
• The sum of the three altitudes of a triangle is less than the sum of the three sides of the triangle.
•  The point of concurrency of the bisector of the angles of a triangle is called incenter.
•  The point of concurrency of the perpendicular bisector of the sides of a triangle is called circumcenter.
•  The point of concurrency of the three altitudes of a triangle is called orthocenter.
•   Median is a line segment joining a vertex of a triangle to the midpoint of its opposite side.
•  The medians of triangle passes through the same point known as centroid of triangle and it divides each median in the ratio of 2:1.
•  Triangles on the same base and between the same parallels are equal in area.
•  A median divides a triangle into two triangles of equal area.
•  If equilateral triangles are drawn on the sides of a right angle triangle, then the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
•  If a perpendicular drawn from the opposite vertex to the hypotenuse of right triangle, the triangles on the opposite sides of the perpendicular are similar to each other.
•  The line segment joining the midpoint of the hypotenuse of a right triangle to the vertex of the right angle is equal to half the hypotenuse.

Theorem about Similarity of triangles

Two triangles are said to be similar if

• If two corresponding internal angles of two triangles have the same measure, the triangles are similar.
• If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
• If three corresponding sides of two triangles are in proportion, then the triangles are similar.

Properties of Similar Triangles

•   If the corresponding sides of two triangles are proportional, then the triangles are similar.
•  The ratio of area of similar triangles is equal to the ratio of the square of the corresponding sides/altitudes/angle bisectors and medians.
•  The ratio of perimeters of similar triangles is equal to the ratio of the square of the corresponding sides/altitudes/angle bisectors and medians.
• Ratio of corresponding altitudes of two similar triangles is equal to the ratio of their corresponding sides.    

Congruence Axiom

1. SAS (Side-Angle-Side) Congruence theorem:

The two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding sides and angle of other triangle.

2. ASA (Angle-Side- Angle) Congruence theorem:

The two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding angles and side of other triangle.

3. SSS (Side-Side-Side) Congruence theorem:

The two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of other triangle.

4. AAA (Angle-Angle- Angle) Congruence theorem:

The two triangles are congruent if three angles of one triangle are equal to the corresponding three angles of other triangle.

5. RHS (Right-Hypotenuse-Side) Congruence theorem:

Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and the corresponding side.

Properties

• The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and  equal to half of it.
• If a line is drawn parallel to one side of a triangle the other two sides are divided in the same ratio.
• If two lines intersect, then vertically opposite angles are equal.
• If one of the four angles formed by two intersecting lines is a right angle, then each of the four angles is a right angle.
• If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
• If a transversal intersects two parallel lines, then each pair of alternate and corresponding angles are equal.
• If a transversal intersect two lines in such a way that a pair of consecutive interior angles are supplementary, then two lines are parallel.
• If two lines are perpendicular to the same line, then they are parallel to each other.
• If two parallel lines are interested by a transversal, then bisectors of any two corresponding angles and alternate angles are parallel.
• Two line segments are congruent if and only if their lengths are equal.
• Two line angles are congruent if and only if their measures are equal.

ü   

Quadrilateral

• In a regular polygon, all sides and interior angles are equal.
• The sum of the interior angles of a quadrilateral is 360 degrees.
• The exterior angle of a regular polygon is 360 degree/n.
• The interior angle of a regular polygon is (n-2/n) x 180 degree.
• The perimeter of a quadrilateral is greater than the sum of diagonals.

1. Parallelogram

A quadrilateral with each pair of opposite sides parallel.

Properties:

ØThe opposite angles are equal.

Ø  The opposite angles are equal.

Ø  The diagonals bisect each other.

2. Rhombus

A quadrilateral with each pair of opposite sides parallel and all the four sides are equal.

Properties:

Ø  The opposite angles are equal.

Ø  The diagonals perpendicularly bisect each other.

3. Rectangle

A quadrilateral with each pair of opposite sides parallel and equal.

Properties:

Ø  All four angles are right angles.

Ø  The diagonals are of equal length.

Ø  The diagonals bisect each other.

4. Square

A quadrilateral with each pair of opposite sides parallel and all the four sides are equal.

Properties:

Ø  All four angles are right angles.

Ø  The diagonals are of equal length.

Ø  The diagonals perpendicularly bisect each other.

5. Trapezium

A quadrilateral with one pair of the opposite sides is parallel.

Property:

The diagonal of a trapezium cut each other in the same ratio.     
    

6. Kite

A quadrilateral with exactly two pair of equal consecutive sides.
Property:

Ø  The diagonal are perpendicular to one another.

Ø  One of the diagonal bisects the other.

Ø  In the figure  < B = < D  But < A not equal to < C

•    The quadrilateral formed by joining the mid points of   consecutive sides of a quadrilateral is a parallelogram.
•    A diagonal of a parallelogram divides it into two triangles of equal area. 
•    Diagonals of a parallelogram divide it into four triangles of equal area.   
•     Any straight line parallel to the parallel sides of a trapezium cuts the other two sides proportionally.

      

Recruitment: Canara bank

Canara Bank

Recruitment of 2000 Clerks Project 2012

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Recruitments : Federal Bank

Federal Bank

Recruitment of Probationary Officers and Clerks Project 2012

We are pleased to invite applications for recruitment staff in our organization,   from candidates who have valid IBPS Score Card of  Common Written Examination (CWE) conducted in 2011. 

Important Dates:

Payment of application fees	8th June 2012-25th June 2012
Opening date for On-line registration	11th June 2012
Last date for On-line registration	25th June 2012

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Line Graph: Import- Export

Line Graph

1.Import-Export Graph

Direction: Study the following line graph carefully and answer the questions.

The following graph shows the ratio of import to export by two companies over the years:

In this graph several cases are there in which answer is cannot be determine.

   

    1.      We cannot relate two companies together unless the further information is not given :

For Example:

•   The export of company A in 2007 was Rs. 325 crore. What was the import of company B in the year 2009?

Answer: Cannot be determine

• The export of company B in 2008 was Rs. 348 crore. What was the import of company A in the same year?

Answer: Cannot be determine

      2 .      We cannot relate two years together unless the further information is not given :

For Example:

•   The import of company A in 2010 was Rs. 75 crore. What was the export of company A in the year 2006?

Answer: Cannot be determine

•   The export of company B in 2007 was Rs. 740 crore. What was the import of company B in the year 2008?

Answer: Cannot be determine

      3.      If the sum of Import/ Export of two companies together is given, then any other term Export/ Import is asked:

For Example:

•   If the total export of company A and company B in year 2010 together is Rs. 85 crore. What will be the total import of both the companies together in the same year?

Answer: Cannot be determine

      4.      If the sum of Import/ Export of  a company in two years together is given, then any other term Export/ Import is asked:

For Example:

•  If the total export of company A in 2007 and 2008 together are Rs. 784 crore. What will be the total import of company A in 2007 and 2008 together?

Answer: Cannot be determine

   

The following questions which answer are determined:

For Example:

Type-I

      1.      If the export of company A in 2009 is Rs. 150 crore. Then what will be the import of company A in 2009?

     Solution: Import of company A in 2009 = 150 x (7/5)
                                                                      = 210 crore

       

    2.      If the import of company B in 2006 is Rs. 240 crore. Then what will be the export of company B in 2006?

Solution: Export of company B in 2006 = 240 x (5/4)
                                                                      = 300 crore
 

    3.    If the import of company A in 2007 is Rs. 375 crore . Then what will be the total export and import of Company A in 2007?

Solution: Total export and import of company A in 2007 = 375 x (13/3)
                                                                      = 1625 crore

 

    4.      If the export of company B in 2009 is Rs. 380 crore . Then what will be the difference between import and export of company B in 2009?

   Solution: Difference between import and export of company B in 2009 = 380 x (1/5)

                                                                      = 76 crore

 

 

Type-II

   5.      In how many of the given years the export of company A is more than the import of company A?
   Solution: In four years the export of company A is more than the import of company A i.e, 2005, 2006, 2007 and 2010.
    Ratio of import to export
    2005 = 3:5
    2006 = 9:10
    2007 = 3:10
    2008 = 6:5
    2009  = 7:5
    2010 = 4:5

    2011 = 1:1

  

    6.      In how many of the given years the export of company B is less than the import of company B?

Solution: In two years the export of company B is less than the import of company B i.e, 2009 and 2010.
Ratio of import to export
    2005 = 1:2

    2006 = 4:5
    2007 = 3:5

    2008 = 1:1

    2009  = 6:5
    2010 = 7:5

        2011 = 4:5

Type-III

7.  If the import of company B in 2006 and the import of company A in 2007 is Rs. 144 crore and Rs. 321 crore respectively, then what is the total export of company B in 2006 and company A in 2007?

Solution: Import of company B in 2006 = 144
                 Export of company B in 2006 = 144 x (5/4)
                                                                  = 180 crore
                 Import of company A in 2007 = 321
                 Export of company A in 2007 = 321 x (10/3)

                                                                      = 1070 crore
                     Total export of company B in 2006 and company A in 2007 = 180 + 1070 = 1250 crore

Line Graph: Profit Percentage

Line Graph

    1.    Profit Percentage Graph

Direction: Study the following line graph carefully and answer the questions.

The following graph shows the percentage profit of two companies:

In this graph several cases are there in which answer is cannot be determine.

   

    1.      We cannot relate two companies together unless the further information is not given :

For Example:

•   The income of company A in 2009 was Rs. 7.65 crore. What was the income of company B in the year 2009?

Answer: Cannot be determine

• The expenditure of company B in 2006 was Rs. 185 crore. What was the income of company A in the same year?

Answer: Cannot be determine

      2 .      We cannot relate two years together unless the further information is not given :

For Example:

•   The income of company B in 2008 was Rs. 83 crore. What was the expenditure of company B in the year 2007?

Answer: Cannot be determine

•   The expenditure of company A in 2007 was Rs. 245 crore. What was the income of company A in the year 2008?

Answer: Cannot be determine

      3.      If the sum of Income/ Expenditure/Profit of two companies together is given, then any other term Expenditure/Profit/ Income is asked:

For Example:

•   If the total expenditure of company A and company B in year 2010 together is Rs. 28 crore. What will be the total income of both the companies together?

Answer: Cannot be determine

      4.      If the sum of Income/ Expenditure/Profit of  a company in two years together is given, then any other term Expenditure/Profit/ Income is asked:

For Example:

•  If the total expenditure of company A in 2007 and 2008 together are Rs. 36 crore. What will be the total income of company A in 2007 and 2008 together?

Answer: Cannot be determine

     5.      Profit related question are asked:

For Example:

•   What is the average profit of Company A over the years?

Answer: Cannot be determine

•  What is the required ratio of both the companies in the year 2008?

Answer: Cannot be determine

•   What is the percentage increase/decrease in profit of Company A in the year 2007 from the previous year?

Answer: Cannot be determine

The following questions which answer are determined:

For Example:

Type-I

      1.      If the expenditure of company A in 2009 is Rs. 150 crore. Then what will be the income of company A in 2009?

              

 

     

    

    2.      If the expenditure of company B in 2006 is Rs. 240 crore. Then what will be the profit of company B in 2006?

 

    

    

    3.      If the income of company A in 2007 is Rs. 675 crore . Then what will be the expenditure of Company A in 2007?

 

   

    

    

     4.      If the income of company B in 2008 is Rs. 560 crore . Then what will be the profit of company B in 2008?

     

 

  

    

    5.      If the profit of company A in 2010 is Rs. 280 crore. Then what will be the expenditure of company A in 2010?

 

   

    

     

    6.      If the profit of company B in 2011 is Rs.350 crore. Then what will be the income of company B in 2011?

Type-II

   7.      The ratio of expenditure of company A and company B is equal in the year 2009. Then what will be the ratio of income in the same year?

Required Ratio = 26:27

   

     

  

    8.      The ratio of income of company A and company B is equal in the year 2007. Then what will be the ratio of expenditure in the same year?

Required Ratio = 25:27

Type-III

   9.      What is the percentage increase in profit % of company B in the year 2008 from the previous year?

Type-IV

10.  If the income of company B in 2007 and the income of company A in 2011 is Rs. 15 crore and Rs. 62 crore respectively, then what is the total expenditure of company B in 2007 and company A in 2011?