CDS preparation - triangle (approx 5-6 questions) formulae and theorems .. - Pk hindi tech :- Technologies se related jaankari hindi me

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CDS preparation - triangle (approx 5-6 questions) formulae and theorems ..

CDS preparation - triangle (approx 5-6 questions) formulae and theorems ..

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CDS ,TRIANGLE 


SOME BASIC TERMS -


**Altitude  and Ortho-centre**:-


-   The ALTITUDE  of a triangle is the line segment perpendicular drawn from the vertex to the side opposite to it. The side On which perpendicular is drawn is called its BASE.






The point of intersection of all the three ALTITUDES of a triangle is called its ORTHOCENTRE.  


** Medians and Centroid*:-





-  A line segment joining the mid- point of a side to the opposite vertex is called MEDIAN.


- The point of intersection of all the three medians of a triangle is called its CENTROID .


- CENTROID divides the median in the ratio of  2:1.


- A MEDIAN bisects the area of the triangle .




** Incentre and Angle bisector**:-





- The point of intersection of all the three angle bisectors of a triangle is called its INCENTRE , denoted by I.


- The circle with centre I and touches all the side is called incircle, and the radius of this circle is called INRADIUS, denoted by *r*.
                                           *Inradius*=1/3 of height= (side)/2 √3



**Perpendicular Bisector and Circumcentre**:-




- The point of intersection of the perpendicular bisectors of the sides of a triangle is called its CIRCUMCENTRE, denoted by C.


- The circle with centre c and passing through vertices A,B and C is called CIRCUMCENTRE . Its radius is called CIRCUMRADIUS,denoted by R.


                      Circumradius= 2/3 of height=( side)/√3 



--** Some important results of a triangle**--



* If two sides of a triangle are equal than the side opposite to them are also equal.


* In a triangle the side opposite to the largest angle is the longest side.


* The sum of all the interior angle of a triangle is 180 degrees.


* If the side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior opposite angles.


*  A triangle must have atleast two acute angles.


*If a perpendicular is drawn from the vertex of a right angled triangle to the hypotenuese , then the triangle on both side of the perpendicular are similar to the original triangle and also to each other.


* A triangle is Isosceles if its two altitudes are equal.


* If the three sides of a triangle are produced then the sum of all the exterior angles so formed is 360 degrees.


* If the bisector of the vertical angle bisects the base then the triangle is an isosceles.


* Medians of an equilateral triagle are equal.


*Perimeter of a triangle is greater than the sum of its three medians.
 


---** Some more important results of a triangle**---


1 -In a right angled triangle ABC, where angle B=90 degree and AC is hypotenuese .The perpendicular BD is dropped on the hypotenuese AC from right angled vertex B,then-
  



  (1) BD=( AB×BC) / AC
 (2) AD=(AB)² /AC
 (3) CD= (BC)² /AC
 (4) 1/(BD)² = 1/(AB)² + 1/(BC)²


2- The area of an equilateral triangle discribed on the side of a square is half of a equilateral triangle discribed on its diagonal.


3- In a triangle ABC if angle B>90 degree and AD is perpendicular drawn on BC, then-



                               (AC)²= (AB)² + (BC)² + 2BC×BD

4-  In a triangle ABC if angle B<90 degree and AD is perpendicular drawn on BC, then-


                                 (AC)²= (AB)² + (BC)² - 2BC×BD

5 - In a triangle ABC ,  three times the sum of square of sides of a triangle is equal to four times the sum of  square of medians of a triangle.



                                      3[(AB)² + (BC)² + (CD)²] = 4 [(AD)² + (CF)² + (BE)²

6-  In a equilateral triangle ABC , the side BC  is trisected at D then,



                           9(AD)² = 7(AC)²


7-  When the bisector of one internal angle and the bisector of external meet at a point then the formed angle is equal to half of the verticle angle.
                      

                     Angle BEC = 1/2 of angle BAC


8- In a triangle ABC the  side BC is produced to D, and the bisector of  angle A  meets BC at L then,


                     angle  ABC + angle ACD = 2 Angle ALC 


9- In a triangle  ABC, the  bisector of  angle  B and angle C meets each other at a point O then,


                 Angle BOC = 90 degree + 1/2 of angle A


10-  In a triangle ABC the side AB and AC are produced to P and Q , the bisector of angle PBC and QCB intersect at point O, then-


        Angle BOC = 90 degree - 1/2 angle A

11- In a triangle ABC , angle B > angle C, If  AN is the bisector of Angle BAC and AM is the perpendicular on BC then,


       Angle MAN = 1/2 ( angle B - angle C)


12- The sum of any two sides of a triangle is greater than twice the median drawn to the third side.

        AB + BC > 2 AD







 --**Congruent triangle**--


 Two triangles are congruent if  both are exactly of same size i.e all angles and sides are equal to corresponding angles and sides of others.



--** Similar Triangles**--


Two triangles are said to be similar if -


a)- Their corresponding sides are propotional.


b)- Their corresponding angles are equal.



---** Some results on Similar triangles**---



(1) Basic Proportionality Theorem-


   If a line parallel to one side of a triangle is drawn which intersects the other two sides in distinct point, then it divides these sides into the same ratio.
  


        In triangle ABC 
                             DE is parallel to BC
       therefore, AD/ DB = AE/ EC
                     or, AD / AB = AE/ AC
                    or, AB / BD= AC/EC

(2)  The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides contaoning the angles.



                         AB / AC = BD / DC
        
(3)  The line joining the mid points of any two sides of a triangle is parallel to the third side and is half of the third side.



            D and E are the mid points 0f AB and AC respectively.
   therefore, DE is parallel to BC and ,
                       
                       DE = 1/2 BC
(4)  If two triangles are equiangular, then the ratio of their corresponding side is the same as the ratio of their corresponding Altitudes.


              BC / QR  = AD / PS
(5)  If two triangles are equiangular, then the ratio of their corresponding side is the same as the ratio of their corresponding angle bisector segments.


              BC / QR  = AD / PS
(6)  If two triangles are equiangular, then the ratio of their corresponding side is the same as the ratio of their corresponding Median.


              BC / QR  = AD / PS
(7)  If two triangles are equiangular, then the ratio of their corresponding side is the same as the ratio of their Perimeter.


 ---** Area of similar triangles**---


 

(1)  The ratio of area of two similar triangles is equal to the ratio of square of their corresponding sides.


        ar. ( triangle ABC) / ar. (triangle PQR) = ( AB)² / (PQ)²
                                                                                   = (AC)² / (PR)²
                                                                                   = ( BC)² / (QR)²

(2)  The ratio of area of two similar triangles is equal to the square of their corresponding altitude.


       ar. ( triangle  ABC) / ar. (triangle PQR) =  (AD)² / (PS)²

(3)  The ratio of area of two similar triangles is equal to the square of their corresponding medians .


      ar. ( triangle  ABC) / ar. (triangle PQR) =  (AD)² / (PS)²
 

(4)  The ratio of area of two similar triangles is equal to the square of their corresponding angle bisector .


      ar. ( triangle  ABC) / ar. (triangle PQR) =  (AD)² / (PS)²     

(5)  If the areas of two similar triangles are equal then the triangles are congruent.
   

  OR
        
     Equal and similar triangles are congruent.

(6)  The line segment joining the mid- points of the sides of a triangle forms four triangles , each of them are similar to the original triangle.
       




           





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