SSC CGL Previous Year Questions (2023 - 18): Geometry | SSC CGL Mathematics Previous Year Paper (Topic-wise) PDF Download

 Page 1


 Pinnacle  Day: 30th - 38th  Geometry 
 Geometry 
 LINES 
 1).  Parallel  Lines  :  Two  or  more  lines 
 which  will  never  meet  just  like  railway 
 track. 
 2).  Transversal  Lines:  A  line  which  cuts 
 parallel  lines  as  shown  in  the  ?gure 
 below. 
 ?  EF  is  the  transverse  line.  AB  and 
 CD  are  parallel  lines.  Symbol  of 
 parallel  lines  is  ‘||’.  To  denote  a 
 line  symbols  like  or  simply  ???? 
 AB are used. 
 ?  Let’s  take  a  look  at  various 
 angles  made  by  the  transverse 
 line and the parallel lines : 
 (i)  Corresponding  Angles:-  and  ,  ?1  ?5  ?2 
 and  ,  and  ,  and  are  pairs  ?6  ?4  ?8  ?3  ?7 
 of corresponding angles. 
 Corresponding  angles  are  equal.  E.g. •
 ?1 = ?5 
 (ii)  Alternate  Angles:-  E.g.  (  ),  (  ?1 = ?7 
 ), (  ),  (  ).  ?2 = ?8  ?4 = ?6  ?3 = ?5 
 (iii)  Vertically opposite angles  :- 
 E.g. (  ), (  ), (  ),  ?1 = ?3  ?2 = ?4  ?6 = ?8 
 (  )  ?5 = ?7 
 (iv)  Adjacent  angles:-  in  ?1 + ?2 = 180° 
 this  case  as  these  are  linear  pairs.  It  is  not 
 necessary that their sum should be  .  180° 
 (v)  Sum  of  Interior  angles  on  same  side  = 
 2×  right  angles  =  .  The  angle,  made  180° 
 by bisectors of interior angle will be  .  90° 
 (vi)  Sum  of  Exterior  angles  on  same  side 
 =  180° 
 Note:  From  a  point,  an  in?nite  number  of 
 lines can be drawn. 
 3).  Internal  angle  bisector  and  External 
 angle  bisector  .  In  the  given  ?gure  AB  is 
 the  internal  angle  bisector  and  AC  is  the 
 external angle bisector. 
 ANGLES 
 Various types of angles are as follows: 
 1).  Acute Angle: Angles which are less 
 than  .  90° 
 2).  Right Angle:  angle is called right  90° 
 angle. 
 3).  Obtuse Angle: Angles greater than 
 and less than  .  90°  180° 
 4).  Straight  Angles:  Angle  equal  to  .  180° 
 5).  Re?ex  angle:  Angles  greater  than 
 and less than  .  180°  360° 
 6).  Complete Angle:  angle is called  360° 
 Complete angle or whole angle. 
 7).  Complementary  Angles:  Sum  of  two 
 angles is equal to  90° .
 8).  Supplementary  Angles:  Sum  of  the 
 two angles is equal to  180° .
 TYPES OF TRIANGLES 
 1). Based on sides:- 
 ?  Equilateral  Triangles:  All  three 
 sides are equal to each other. 
 ?  Isosceles  Triangles  :  Any  two 
 sides are equal to each other. 
 ?  Scalene  Triangle:  All  three  sides 
 are different in length. 
 2). Based on angles :- 
 ?  Right  angle  triangle  :-  One  of  the  angle 
 is  90° .
 ?  Obtuse  angled  triangle  :-  One  angle  is 
 more than  90° .
 ?  Acute  angled  triangle  :-  All  three 
 angles are less than  90° .
 Properties of Triangles 
 [1.]  If  a  line  DE  intersects  two  sides  of  a 
 triangle  AB  and  AC  at  D  and  E 
 respectively,  then  :  =  and 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 =  .  Also  if  D  and  E  are  mid  points  of 
 ???? 
 ???? 
 AB and AC respectively, then: 
 =  and DE will be parallel to BC.  ???? 
 ???? 
 2 
 E.g.  Let  AD  =  2.5  units;  DB  =  5,  AE  =  2  and 
 BC = 9, then EC = ? and DE = ? 
 =  = 
 ???? 
 ???? 
 ???? 
 ???? 
? ???? = ????     ×    
 ???? 
 ???? 
 2     ×    
 5 
 2 . 5 
 =  Similarly DE = 3 units.  4     ?????????? .
 [2.]  In  case  of  Internal  angle  bisector: 
 or, 
 ???? 
 ???? 
=
 ???? 
 ???? 
 ???? 
 ???? 
=
 ???? 
 ???? 
 Here  PS  is  the  internal  angle  bisector  and 
 a  common  side  of  the  triangles 
 .  ?     ??????     ??????     ?  ?????? 
 [3.]  In case of  Exterior angle bisector : 
 = 
 ???? 
 ???? 
 ???? 
 ???? 
 PS is the external angle bisector. 
 Congruency of Triangles 
 Two triangles will be congruent if: 
 1.  SSS:  (Side  –  Side  –  Side  rule):  When 
 all three sides are equal. 
 2.  SAS:  (Side  –  Angle  –  Side  rule):  When 
 two  sides  and  one  angle  are  equal. 
 3.  ASA:  (Angle  –  Side  –  Angle  rule): 
 When two angles and one side are equal. 
 4.  RHS:  (Right  angle  –  Hypotenuse  – 
 Side  rule):  When  one  side  and 
 hypotenuse  of  the  right  angled  triangle 
 are equal. 
 Similarity of Triangles 
 1).  Similar  triangles  are  triangles  that 
 have the same shape, but their sizes may 
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Page 2


 Pinnacle  Day: 30th - 38th  Geometry 
 Geometry 
 LINES 
 1).  Parallel  Lines  :  Two  or  more  lines 
 which  will  never  meet  just  like  railway 
 track. 
 2).  Transversal  Lines:  A  line  which  cuts 
 parallel  lines  as  shown  in  the  ?gure 
 below. 
 ?  EF  is  the  transverse  line.  AB  and 
 CD  are  parallel  lines.  Symbol  of 
 parallel  lines  is  ‘||’.  To  denote  a 
 line  symbols  like  or  simply  ???? 
 AB are used. 
 ?  Let’s  take  a  look  at  various 
 angles  made  by  the  transverse 
 line and the parallel lines : 
 (i)  Corresponding  Angles:-  and  ,  ?1  ?5  ?2 
 and  ,  and  ,  and  are  pairs  ?6  ?4  ?8  ?3  ?7 
 of corresponding angles. 
 Corresponding  angles  are  equal.  E.g. •
 ?1 = ?5 
 (ii)  Alternate  Angles:-  E.g.  (  ),  (  ?1 = ?7 
 ), (  ),  (  ).  ?2 = ?8  ?4 = ?6  ?3 = ?5 
 (iii)  Vertically opposite angles  :- 
 E.g. (  ), (  ), (  ),  ?1 = ?3  ?2 = ?4  ?6 = ?8 
 (  )  ?5 = ?7 
 (iv)  Adjacent  angles:-  in  ?1 + ?2 = 180° 
 this  case  as  these  are  linear  pairs.  It  is  not 
 necessary that their sum should be  .  180° 
 (v)  Sum  of  Interior  angles  on  same  side  = 
 2×  right  angles  =  .  The  angle,  made  180° 
 by bisectors of interior angle will be  .  90° 
 (vi)  Sum  of  Exterior  angles  on  same  side 
 =  180° 
 Note:  From  a  point,  an  in?nite  number  of 
 lines can be drawn. 
 3).  Internal  angle  bisector  and  External 
 angle  bisector  .  In  the  given  ?gure  AB  is 
 the  internal  angle  bisector  and  AC  is  the 
 external angle bisector. 
 ANGLES 
 Various types of angles are as follows: 
 1).  Acute Angle: Angles which are less 
 than  .  90° 
 2).  Right Angle:  angle is called right  90° 
 angle. 
 3).  Obtuse Angle: Angles greater than 
 and less than  .  90°  180° 
 4).  Straight  Angles:  Angle  equal  to  .  180° 
 5).  Re?ex  angle:  Angles  greater  than 
 and less than  .  180°  360° 
 6).  Complete Angle:  angle is called  360° 
 Complete angle or whole angle. 
 7).  Complementary  Angles:  Sum  of  two 
 angles is equal to  90° .
 8).  Supplementary  Angles:  Sum  of  the 
 two angles is equal to  180° .
 TYPES OF TRIANGLES 
 1). Based on sides:- 
 ?  Equilateral  Triangles:  All  three 
 sides are equal to each other. 
 ?  Isosceles  Triangles  :  Any  two 
 sides are equal to each other. 
 ?  Scalene  Triangle:  All  three  sides 
 are different in length. 
 2). Based on angles :- 
 ?  Right  angle  triangle  :-  One  of  the  angle 
 is  90° .
 ?  Obtuse  angled  triangle  :-  One  angle  is 
 more than  90° .
 ?  Acute  angled  triangle  :-  All  three 
 angles are less than  90° .
 Properties of Triangles 
 [1.]  If  a  line  DE  intersects  two  sides  of  a 
 triangle  AB  and  AC  at  D  and  E 
 respectively,  then  :  =  and 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 =  .  Also  if  D  and  E  are  mid  points  of 
 ???? 
 ???? 
 AB and AC respectively, then: 
 =  and DE will be parallel to BC.  ???? 
 ???? 
 2 
 E.g.  Let  AD  =  2.5  units;  DB  =  5,  AE  =  2  and 
 BC = 9, then EC = ? and DE = ? 
 =  = 
 ???? 
 ???? 
 ???? 
 ???? 
? ???? = ????     ×    
 ???? 
 ???? 
 2     ×    
 5 
 2 . 5 
 =  Similarly DE = 3 units.  4     ?????????? .
 [2.]  In  case  of  Internal  angle  bisector: 
 or, 
 ???? 
 ???? 
=
 ???? 
 ???? 
 ???? 
 ???? 
=
 ???? 
 ???? 
 Here  PS  is  the  internal  angle  bisector  and 
 a  common  side  of  the  triangles 
 .  ?     ??????     ??????     ?  ?????? 
 [3.]  In case of  Exterior angle bisector : 
 = 
 ???? 
 ???? 
 ???? 
 ???? 
 PS is the external angle bisector. 
 Congruency of Triangles 
 Two triangles will be congruent if: 
 1.  SSS:  (Side  –  Side  –  Side  rule):  When 
 all three sides are equal. 
 2.  SAS:  (Side  –  Angle  –  Side  rule):  When 
 two  sides  and  one  angle  are  equal. 
 3.  ASA:  (Angle  –  Side  –  Angle  rule): 
 When two angles and one side are equal. 
 4.  RHS:  (Right  angle  –  Hypotenuse  – 
 Side  rule):  When  one  side  and 
 hypotenuse  of  the  right  angled  triangle 
 are equal. 
 Similarity of Triangles 
 1).  Similar  triangles  are  triangles  that 
 have the same shape, but their sizes may 
 www.ssccglpinnacle.com                                                 Download Pinnacle Exam Preparation App 185
 Pinnacle  Day: 30th - 38th  Geometry 
 vary. 
 2).  In  congruency  the  triangles  are  mirror 
 images  of  each  other.  We  can  say  that  all 
 congruent  triangles  are  similar  but  the 
 vice versa is not true. 
 3).  Important  properties  of  similar 
 triangles: 
 ?  = 
 ????????     ????     ?  ?????? 
 ????????     ????     ?  ?????? 
 ???? 
 2 
 ???? 
 2 
 = 
 ???? 
 2 
 ???? 
 2 
   =   
 ???? 
 2 
 ???? 
 2 
 ?  = 
 ??????????????????     ????     ?  ?????? 
 ??????????????????     ????     ?  ?????? 
 ???? 
 ???? 
=
 ???? 
 ???? 
   =   
 ???? 
 ???? 
 Altitude : 
 ?  It  is  also  known  as  height.  The 
 line  segment  drawn  from  the 
 vertex  of  a  triangle 
 perpendicular  to  its  opposite 
 side  is  called  an  Altitude  of  a 
 triangle. 
 Orthocenter: 
 ?  The  point  at  which  the  altitudes 
 of  a  triangle  intersect  is  called 
 an Orthocenter. 
 ?  Generally  Orthocenter  is 
 denoted by ‘O’. 
 ?  Here,  AOC = 180° -  B  ?  ? 
 ?  In  the  case  of  a  right  angled 
 triangle,  the  orthocenter  lies  at 
 the vertex of  right angle. 
 ?  In  case  of  an  obtuse  angled 
 triangle,  the  orthocenter  lies 
 outside the triangle. 
 In-center: 
 The  point  at  which  the  internal  angle 
 bisectors  of  a  triangle  meet.  It  is 
 generally  denoted  by  ‘I’.  From  the 
 in-center  the  in-circle  of  a  triangle  is 
 drawn.  The  radius  of  the  in-circle  is  equal 
 to ID = IE = IF. 
 Important Results: 
 ?  +  (for  Internal  ?  ?????? = 90° 
 ?  ?? 
 2 
 angle bisectors) 
 ?  (for  ?  ?????? = 90° -   
 ?  ?? 
 2 
 external angle bisectors) 
 ?  = 
 ???? 
 ???? 
 ???? 
 ???? 
 ?  In-radius r  = 
 ????????     ????     ???????????????? 
 ???????? - ??????????????????     ????     ???????????????? 
 ?  AI : ID = AB + AC : BC 
 BI : IE = AB + BC : AC 
 CI : IF = AC + BC : AB 
 Length of angle bisector(AD)    •
 = 
 2     ×     ????        ×     ????     ?????? ?
 ????       +       ???? 
 Circum-center :- 
 The  point  at  which  the  perpendicular 
 bisectors  of  the  sides  of  the  triangle 
 meet. 
 ?  From  the  circum-center  the 
 circumcircle  of  a  triangle  is 
 drawn.  The  radius  of  the 
 circum-circle  is  equal  to  PG  = 
 RG = QG. 
 ?  Since  the  angle  formed  at  the 
 center  of  a  circle  is  double  of 
 the  angle  formed  at  the 
 circumference, we have:: 
 ,  ?  ?????? = 2?  ??  ?  ?????? = 2?  ?? 
 and  .     ?  ?????? = 2?  ?? 
 ?  For  an  obtuse  angle  triangle,  the 
 circumcenter  lies  outside  the 
 triangle. 
 Note:  A  line  which  is  perpendicular  to 
 another  line  and  also  bisects  it  into  two 
 equal  parts  is  called  a  perpendicular 
 bisector. 
 Centroid:  It  is  the  point  of  intersection  of 
 the  medians  of  a  triangle.  It  is  also  called 
 the center of mass. 
 ?  The  median  is  a  line  drawn  from 
 the  vertex  of  a  triangle  to  the 
 mid  –  point  of  the  opposite 
 side. 
 ?  Each  median  is  divided  in  the 
 ratio of 2 : 1 at the centroid. 
 ?  Median  divides  the  triangle  into 
 two equal areas. 
 ?  Apollonius theorem: 
 ? 
 ???? 
 2 
+ ???? 
 2 
= 2 ( ???? 
 2 
+ ???? 
 2 
)
 Other key points for medians: 
 ?  All  the  medians  of  a  triangle 
 divide  it  into  six  parts  with  equal 
 areas. 
 Therefore,  ????????     ????     ?  ?????? 
= 6  ×  ????????  ????  ??????  ????  ??h?? 
    ??????????????     ?????????? .
 ?  ????????     ????     ?  ?????? 
 = 
 1 
 12 
    ×     ????????     ????  ?  ?????? 
 ?  FE  divides  the  line  AG  in  the 
 ratio of 3 : 1 i.e.  . 
 ???? 
 ???? 
=
 3 
 1 
 ?  Area  of  ABG  =  Area  of  BCG  = ? ?
 Area of  ACG ?
 ?  Area of  DEF =  ABC ?
 1 
 4 
?
 Note  :  In  case  of  Equilateral  triangle, 
 orthocenter,  in-center,  circum-center  and 
 centroid lie at the same point. 
 Other Facts about triangles: 
 1).  Sum of two sides is always greater 
 than the third side. 
 E.g.  AB  +  BC  >  CA,  AB  +  AC  >  BC,  BC  +  CA 
 > AB. 
 2).  Difference of two sides is always less 
 than the third side. 
 E.g.  AB  -  BC  <  CA,  AB  -  AC  <  BC,  BC  -  CA 
 < AB. 
 3).  In  ABC,  B  >  C,  if  AM  is  the ? ?  ? 
 bisector of  BAC and AN  BC, then  ? ?
 MAN =  (  B -  C)  ? 
 1 
 2 
 ?  ? 
 4).  In  right  angled  ABC,  where  BD  is  an ?
 altitude  to  hypotenuse  AC,  then  Altitude 
 BD =  . 
 ????    ×    ???? 
 ???? 
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Page 3


 Pinnacle  Day: 30th - 38th  Geometry 
 Geometry 
 LINES 
 1).  Parallel  Lines  :  Two  or  more  lines 
 which  will  never  meet  just  like  railway 
 track. 
 2).  Transversal  Lines:  A  line  which  cuts 
 parallel  lines  as  shown  in  the  ?gure 
 below. 
 ?  EF  is  the  transverse  line.  AB  and 
 CD  are  parallel  lines.  Symbol  of 
 parallel  lines  is  ‘||’.  To  denote  a 
 line  symbols  like  or  simply  ???? 
 AB are used. 
 ?  Let’s  take  a  look  at  various 
 angles  made  by  the  transverse 
 line and the parallel lines : 
 (i)  Corresponding  Angles:-  and  ,  ?1  ?5  ?2 
 and  ,  and  ,  and  are  pairs  ?6  ?4  ?8  ?3  ?7 
 of corresponding angles. 
 Corresponding  angles  are  equal.  E.g. •
 ?1 = ?5 
 (ii)  Alternate  Angles:-  E.g.  (  ),  (  ?1 = ?7 
 ), (  ),  (  ).  ?2 = ?8  ?4 = ?6  ?3 = ?5 
 (iii)  Vertically opposite angles  :- 
 E.g. (  ), (  ), (  ),  ?1 = ?3  ?2 = ?4  ?6 = ?8 
 (  )  ?5 = ?7 
 (iv)  Adjacent  angles:-  in  ?1 + ?2 = 180° 
 this  case  as  these  are  linear  pairs.  It  is  not 
 necessary that their sum should be  .  180° 
 (v)  Sum  of  Interior  angles  on  same  side  = 
 2×  right  angles  =  .  The  angle,  made  180° 
 by bisectors of interior angle will be  .  90° 
 (vi)  Sum  of  Exterior  angles  on  same  side 
 =  180° 
 Note:  From  a  point,  an  in?nite  number  of 
 lines can be drawn. 
 3).  Internal  angle  bisector  and  External 
 angle  bisector  .  In  the  given  ?gure  AB  is 
 the  internal  angle  bisector  and  AC  is  the 
 external angle bisector. 
 ANGLES 
 Various types of angles are as follows: 
 1).  Acute Angle: Angles which are less 
 than  .  90° 
 2).  Right Angle:  angle is called right  90° 
 angle. 
 3).  Obtuse Angle: Angles greater than 
 and less than  .  90°  180° 
 4).  Straight  Angles:  Angle  equal  to  .  180° 
 5).  Re?ex  angle:  Angles  greater  than 
 and less than  .  180°  360° 
 6).  Complete Angle:  angle is called  360° 
 Complete angle or whole angle. 
 7).  Complementary  Angles:  Sum  of  two 
 angles is equal to  90° .
 8).  Supplementary  Angles:  Sum  of  the 
 two angles is equal to  180° .
 TYPES OF TRIANGLES 
 1). Based on sides:- 
 ?  Equilateral  Triangles:  All  three 
 sides are equal to each other. 
 ?  Isosceles  Triangles  :  Any  two 
 sides are equal to each other. 
 ?  Scalene  Triangle:  All  three  sides 
 are different in length. 
 2). Based on angles :- 
 ?  Right  angle  triangle  :-  One  of  the  angle 
 is  90° .
 ?  Obtuse  angled  triangle  :-  One  angle  is 
 more than  90° .
 ?  Acute  angled  triangle  :-  All  three 
 angles are less than  90° .
 Properties of Triangles 
 [1.]  If  a  line  DE  intersects  two  sides  of  a 
 triangle  AB  and  AC  at  D  and  E 
 respectively,  then  :  =  and 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 =  .  Also  if  D  and  E  are  mid  points  of 
 ???? 
 ???? 
 AB and AC respectively, then: 
 =  and DE will be parallel to BC.  ???? 
 ???? 
 2 
 E.g.  Let  AD  =  2.5  units;  DB  =  5,  AE  =  2  and 
 BC = 9, then EC = ? and DE = ? 
 =  = 
 ???? 
 ???? 
 ???? 
 ???? 
? ???? = ????     ×    
 ???? 
 ???? 
 2     ×    
 5 
 2 . 5 
 =  Similarly DE = 3 units.  4     ?????????? .
 [2.]  In  case  of  Internal  angle  bisector: 
 or, 
 ???? 
 ???? 
=
 ???? 
 ???? 
 ???? 
 ???? 
=
 ???? 
 ???? 
 Here  PS  is  the  internal  angle  bisector  and 
 a  common  side  of  the  triangles 
 .  ?     ??????     ??????     ?  ?????? 
 [3.]  In case of  Exterior angle bisector : 
 = 
 ???? 
 ???? 
 ???? 
 ???? 
 PS is the external angle bisector. 
 Congruency of Triangles 
 Two triangles will be congruent if: 
 1.  SSS:  (Side  –  Side  –  Side  rule):  When 
 all three sides are equal. 
 2.  SAS:  (Side  –  Angle  –  Side  rule):  When 
 two  sides  and  one  angle  are  equal. 
 3.  ASA:  (Angle  –  Side  –  Angle  rule): 
 When two angles and one side are equal. 
 4.  RHS:  (Right  angle  –  Hypotenuse  – 
 Side  rule):  When  one  side  and 
 hypotenuse  of  the  right  angled  triangle 
 are equal. 
 Similarity of Triangles 
 1).  Similar  triangles  are  triangles  that 
 have the same shape, but their sizes may 
 www.ssccglpinnacle.com                                                 Download Pinnacle Exam Preparation App 185
 Pinnacle  Day: 30th - 38th  Geometry 
 vary. 
 2).  In  congruency  the  triangles  are  mirror 
 images  of  each  other.  We  can  say  that  all 
 congruent  triangles  are  similar  but  the 
 vice versa is not true. 
 3).  Important  properties  of  similar 
 triangles: 
 ?  = 
 ????????     ????     ?  ?????? 
 ????????     ????     ?  ?????? 
 ???? 
 2 
 ???? 
 2 
 = 
 ???? 
 2 
 ???? 
 2 
   =   
 ???? 
 2 
 ???? 
 2 
 ?  = 
 ??????????????????     ????     ?  ?????? 
 ??????????????????     ????     ?  ?????? 
 ???? 
 ???? 
=
 ???? 
 ???? 
   =   
 ???? 
 ???? 
 Altitude : 
 ?  It  is  also  known  as  height.  The 
 line  segment  drawn  from  the 
 vertex  of  a  triangle 
 perpendicular  to  its  opposite 
 side  is  called  an  Altitude  of  a 
 triangle. 
 Orthocenter: 
 ?  The  point  at  which  the  altitudes 
 of  a  triangle  intersect  is  called 
 an Orthocenter. 
 ?  Generally  Orthocenter  is 
 denoted by ‘O’. 
 ?  Here,  AOC = 180° -  B  ?  ? 
 ?  In  the  case  of  a  right  angled 
 triangle,  the  orthocenter  lies  at 
 the vertex of  right angle. 
 ?  In  case  of  an  obtuse  angled 
 triangle,  the  orthocenter  lies 
 outside the triangle. 
 In-center: 
 The  point  at  which  the  internal  angle 
 bisectors  of  a  triangle  meet.  It  is 
 generally  denoted  by  ‘I’.  From  the 
 in-center  the  in-circle  of  a  triangle  is 
 drawn.  The  radius  of  the  in-circle  is  equal 
 to ID = IE = IF. 
 Important Results: 
 ?  +  (for  Internal  ?  ?????? = 90° 
 ?  ?? 
 2 
 angle bisectors) 
 ?  (for  ?  ?????? = 90° -   
 ?  ?? 
 2 
 external angle bisectors) 
 ?  = 
 ???? 
 ???? 
 ???? 
 ???? 
 ?  In-radius r  = 
 ????????     ????     ???????????????? 
 ???????? - ??????????????????     ????     ???????????????? 
 ?  AI : ID = AB + AC : BC 
 BI : IE = AB + BC : AC 
 CI : IF = AC + BC : AB 
 Length of angle bisector(AD)    •
 = 
 2     ×     ????        ×     ????     ?????? ?
 ????       +       ???? 
 Circum-center :- 
 The  point  at  which  the  perpendicular 
 bisectors  of  the  sides  of  the  triangle 
 meet. 
 ?  From  the  circum-center  the 
 circumcircle  of  a  triangle  is 
 drawn.  The  radius  of  the 
 circum-circle  is  equal  to  PG  = 
 RG = QG. 
 ?  Since  the  angle  formed  at  the 
 center  of  a  circle  is  double  of 
 the  angle  formed  at  the 
 circumference, we have:: 
 ,  ?  ?????? = 2?  ??  ?  ?????? = 2?  ?? 
 and  .     ?  ?????? = 2?  ?? 
 ?  For  an  obtuse  angle  triangle,  the 
 circumcenter  lies  outside  the 
 triangle. 
 Note:  A  line  which  is  perpendicular  to 
 another  line  and  also  bisects  it  into  two 
 equal  parts  is  called  a  perpendicular 
 bisector. 
 Centroid:  It  is  the  point  of  intersection  of 
 the  medians  of  a  triangle.  It  is  also  called 
 the center of mass. 
 ?  The  median  is  a  line  drawn  from 
 the  vertex  of  a  triangle  to  the 
 mid  –  point  of  the  opposite 
 side. 
 ?  Each  median  is  divided  in  the 
 ratio of 2 : 1 at the centroid. 
 ?  Median  divides  the  triangle  into 
 two equal areas. 
 ?  Apollonius theorem: 
 ? 
 ???? 
 2 
+ ???? 
 2 
= 2 ( ???? 
 2 
+ ???? 
 2 
)
 Other key points for medians: 
 ?  All  the  medians  of  a  triangle 
 divide  it  into  six  parts  with  equal 
 areas. 
 Therefore,  ????????     ????     ?  ?????? 
= 6  ×  ????????  ????  ??????  ????  ??h?? 
    ??????????????     ?????????? .
 ?  ????????     ????     ?  ?????? 
 = 
 1 
 12 
    ×     ????????     ????  ?  ?????? 
 ?  FE  divides  the  line  AG  in  the 
 ratio of 3 : 1 i.e.  . 
 ???? 
 ???? 
=
 3 
 1 
 ?  Area  of  ABG  =  Area  of  BCG  = ? ?
 Area of  ACG ?
 ?  Area of  DEF =  ABC ?
 1 
 4 
?
 Note  :  In  case  of  Equilateral  triangle, 
 orthocenter,  in-center,  circum-center  and 
 centroid lie at the same point. 
 Other Facts about triangles: 
 1).  Sum of two sides is always greater 
 than the third side. 
 E.g.  AB  +  BC  >  CA,  AB  +  AC  >  BC,  BC  +  CA 
 > AB. 
 2).  Difference of two sides is always less 
 than the third side. 
 E.g.  AB  -  BC  <  CA,  AB  -  AC  <  BC,  BC  -  CA 
 < AB. 
 3).  In  ABC,  B  >  C,  if  AM  is  the ? ?  ? 
 bisector of  BAC and AN  BC, then  ? ?
 MAN =  (  B -  C)  ? 
 1 
 2 
 ?  ? 
 4).  In  right  angled  ABC,  where  BD  is  an ?
 altitude  to  hypotenuse  AC,  then  Altitude 
 BD =  . 
 ????    ×    ???? 
 ???? 
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 Similarly, if BE is a median drawn to 
 hypotenuse AC, then BE =  AC. (Length 
 1 
 2 
 of  median  is  half  the  length  of 
 hypotenuse) 
 5).  In  right  angled  ABC,  AD  is ?
 perpendicular to BC, then 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 Formulas  of triangle :- 
 In case of equilateral triangle, 
 Area(A) =  (side) 
 2 
•
 3 
 4 
 Inradius(r) = •
 ????????    
 2  3 
 circumradius(R) = •
 ????????    
 3 
 height(h) =  × side •
 3 
 2 
 I  n case of isosceles triangle, 
 Area(A) = •
 ???? 
 4 
 4  ???? 
 2 
- ???? 
 2 
 Where AB = AC 
 Inradius(r) = •
 ????????     ????     ????????????????    
 ???????? - ?????????????????? 
 circumradius(R) = •
 ????     ×     ????     ×     ???? 
 4     ×     ????????     ????     ????????????????    
 height(h) = • ???? 
 2 
-
 ???? 
 2 
 4 
 In case of scalene triangle, 
 Area(A) = • ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 Where a,b,c are sides of triangle 
 s = 
 ??    +    ??    +    ?? 
 2 
 In case of right angle triangle , 
 Area(A) =  × Base × height •
 1 
 2 
 Inradius(r) = •
 ??????????????????????????    +    ????????    -    h?????????????????? 
 2 
 circumradius(R) = •
 h?????????????????? 
 2 
 Note :-  Distance between circumcentre 
 and incentre of any triangle  = 
 ?? 
 2 
   -    2  ???? 
 Sine rule :- 
 =  =  =  2R 
 ?? 
 ???????? 
 ?? 
 ???????? 
 ?? 
 ???????? 
 Where R =  circumradius of triangle 
 Cosine rule :- 
 CosA = 
 ?? 
 2 
+ ?? 
 2 
-    ?? 
 2 
 2  ???? 
 CosB = 
 ?? 
 2 
+    ?? 
 2 
-    ?? 
 2 
 2  ???? 
 CosC = 
 ?? 
 2 
   +    ?? 
 2 
-    ?? 
 2 
 2  ???? 
 If  P  is  any  point  inside  the  equilateral •
 triangle  ,  the  sum  of  altitude  from  the 
 point  P  to  the  sides  AB,  BC  and  AC  equal 
 to the sides of the triangle. 
 +  +  =  a  h 
 1 
 h 
 2 
 h 
 3 
 3 
 2 
 Where a = side of equilateral triangle 
 if  ABC is right angle triangle and AD, • ?
 CE are medians then: 
 (i)  4(AD 
 2 
 + CE 
 2 
 ) = 5 AC 
 2 
 (ii)  AD 
 2 
 + CE 
 2 
 = AC 
 2 
 + ED 
 2 
 If BE and CD are medians and •
 perpendicular to each other then: 
 AB 
 2 
 + AC 
 2 
 = 5 BC 
 2 
 If length of median of triangles are •
 given then area of triangle =  × area of 
 4 
 3 
 triangle formed by medians. 
 CIRCLE 
 Properties of circles :- 
 1).  Angles  in  the  same  segment  of  a 
 circle  from  the  same  base  are  always 
 equal. 
 2).  Angle  dropped  on  the  circumference 
 of  a  circle  with  the  diameter  as  base  is 
 always a right angle. 
 3).  Two  chords  AB  and  CD  of  a  circle 
 intersect,  internally  or  externally,  at  E 
 then: 
 ????  ×  ???? = ????  ×  ???? 
 4).  The  tangent  to  a  circle  at  a  point  on 
 its  circumference  is  perpendicular  to  the 
 radius at that point. 
 OI  PT ?
 5).  Two  tangents  PA  and  PB  are  drawn 
 from  an  external  point  P  on  a  circle  with 
 center O is equal.  PA = PB 
 6).  For  a  tangent,  PT,  and  a  secant,  PB, 
 drawn  to  a  circle  from  a  point  P ,  we  have. 
 ???? 
 2 
= ????  ×  ???? 
 7).  Angle  made  by  a  chord  with  a  tangent 
 is  always  equal  to  the  angle  dropped  on 
 any  point  of  circumference  taking  the 
 chord as base. 
 ?  ?????? = ?  ?????? 
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Page 4


 Pinnacle  Day: 30th - 38th  Geometry 
 Geometry 
 LINES 
 1).  Parallel  Lines  :  Two  or  more  lines 
 which  will  never  meet  just  like  railway 
 track. 
 2).  Transversal  Lines:  A  line  which  cuts 
 parallel  lines  as  shown  in  the  ?gure 
 below. 
 ?  EF  is  the  transverse  line.  AB  and 
 CD  are  parallel  lines.  Symbol  of 
 parallel  lines  is  ‘||’.  To  denote  a 
 line  symbols  like  or  simply  ???? 
 AB are used. 
 ?  Let’s  take  a  look  at  various 
 angles  made  by  the  transverse 
 line and the parallel lines : 
 (i)  Corresponding  Angles:-  and  ,  ?1  ?5  ?2 
 and  ,  and  ,  and  are  pairs  ?6  ?4  ?8  ?3  ?7 
 of corresponding angles. 
 Corresponding  angles  are  equal.  E.g. •
 ?1 = ?5 
 (ii)  Alternate  Angles:-  E.g.  (  ),  (  ?1 = ?7 
 ), (  ),  (  ).  ?2 = ?8  ?4 = ?6  ?3 = ?5 
 (iii)  Vertically opposite angles  :- 
 E.g. (  ), (  ), (  ),  ?1 = ?3  ?2 = ?4  ?6 = ?8 
 (  )  ?5 = ?7 
 (iv)  Adjacent  angles:-  in  ?1 + ?2 = 180° 
 this  case  as  these  are  linear  pairs.  It  is  not 
 necessary that their sum should be  .  180° 
 (v)  Sum  of  Interior  angles  on  same  side  = 
 2×  right  angles  =  .  The  angle,  made  180° 
 by bisectors of interior angle will be  .  90° 
 (vi)  Sum  of  Exterior  angles  on  same  side 
 =  180° 
 Note:  From  a  point,  an  in?nite  number  of 
 lines can be drawn. 
 3).  Internal  angle  bisector  and  External 
 angle  bisector  .  In  the  given  ?gure  AB  is 
 the  internal  angle  bisector  and  AC  is  the 
 external angle bisector. 
 ANGLES 
 Various types of angles are as follows: 
 1).  Acute Angle: Angles which are less 
 than  .  90° 
 2).  Right Angle:  angle is called right  90° 
 angle. 
 3).  Obtuse Angle: Angles greater than 
 and less than  .  90°  180° 
 4).  Straight  Angles:  Angle  equal  to  .  180° 
 5).  Re?ex  angle:  Angles  greater  than 
 and less than  .  180°  360° 
 6).  Complete Angle:  angle is called  360° 
 Complete angle or whole angle. 
 7).  Complementary  Angles:  Sum  of  two 
 angles is equal to  90° .
 8).  Supplementary  Angles:  Sum  of  the 
 two angles is equal to  180° .
 TYPES OF TRIANGLES 
 1). Based on sides:- 
 ?  Equilateral  Triangles:  All  three 
 sides are equal to each other. 
 ?  Isosceles  Triangles  :  Any  two 
 sides are equal to each other. 
 ?  Scalene  Triangle:  All  three  sides 
 are different in length. 
 2). Based on angles :- 
 ?  Right  angle  triangle  :-  One  of  the  angle 
 is  90° .
 ?  Obtuse  angled  triangle  :-  One  angle  is 
 more than  90° .
 ?  Acute  angled  triangle  :-  All  three 
 angles are less than  90° .
 Properties of Triangles 
 [1.]  If  a  line  DE  intersects  two  sides  of  a 
 triangle  AB  and  AC  at  D  and  E 
 respectively,  then  :  =  and 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 =  .  Also  if  D  and  E  are  mid  points  of 
 ???? 
 ???? 
 AB and AC respectively, then: 
 =  and DE will be parallel to BC.  ???? 
 ???? 
 2 
 E.g.  Let  AD  =  2.5  units;  DB  =  5,  AE  =  2  and 
 BC = 9, then EC = ? and DE = ? 
 =  = 
 ???? 
 ???? 
 ???? 
 ???? 
? ???? = ????     ×    
 ???? 
 ???? 
 2     ×    
 5 
 2 . 5 
 =  Similarly DE = 3 units.  4     ?????????? .
 [2.]  In  case  of  Internal  angle  bisector: 
 or, 
 ???? 
 ???? 
=
 ???? 
 ???? 
 ???? 
 ???? 
=
 ???? 
 ???? 
 Here  PS  is  the  internal  angle  bisector  and 
 a  common  side  of  the  triangles 
 .  ?     ??????     ??????     ?  ?????? 
 [3.]  In case of  Exterior angle bisector : 
 = 
 ???? 
 ???? 
 ???? 
 ???? 
 PS is the external angle bisector. 
 Congruency of Triangles 
 Two triangles will be congruent if: 
 1.  SSS:  (Side  –  Side  –  Side  rule):  When 
 all three sides are equal. 
 2.  SAS:  (Side  –  Angle  –  Side  rule):  When 
 two  sides  and  one  angle  are  equal. 
 3.  ASA:  (Angle  –  Side  –  Angle  rule): 
 When two angles and one side are equal. 
 4.  RHS:  (Right  angle  –  Hypotenuse  – 
 Side  rule):  When  one  side  and 
 hypotenuse  of  the  right  angled  triangle 
 are equal. 
 Similarity of Triangles 
 1).  Similar  triangles  are  triangles  that 
 have the same shape, but their sizes may 
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 vary. 
 2).  In  congruency  the  triangles  are  mirror 
 images  of  each  other.  We  can  say  that  all 
 congruent  triangles  are  similar  but  the 
 vice versa is not true. 
 3).  Important  properties  of  similar 
 triangles: 
 ?  = 
 ????????     ????     ?  ?????? 
 ????????     ????     ?  ?????? 
 ???? 
 2 
 ???? 
 2 
 = 
 ???? 
 2 
 ???? 
 2 
   =   
 ???? 
 2 
 ???? 
 2 
 ?  = 
 ??????????????????     ????     ?  ?????? 
 ??????????????????     ????     ?  ?????? 
 ???? 
 ???? 
=
 ???? 
 ???? 
   =   
 ???? 
 ???? 
 Altitude : 
 ?  It  is  also  known  as  height.  The 
 line  segment  drawn  from  the 
 vertex  of  a  triangle 
 perpendicular  to  its  opposite 
 side  is  called  an  Altitude  of  a 
 triangle. 
 Orthocenter: 
 ?  The  point  at  which  the  altitudes 
 of  a  triangle  intersect  is  called 
 an Orthocenter. 
 ?  Generally  Orthocenter  is 
 denoted by ‘O’. 
 ?  Here,  AOC = 180° -  B  ?  ? 
 ?  In  the  case  of  a  right  angled 
 triangle,  the  orthocenter  lies  at 
 the vertex of  right angle. 
 ?  In  case  of  an  obtuse  angled 
 triangle,  the  orthocenter  lies 
 outside the triangle. 
 In-center: 
 The  point  at  which  the  internal  angle 
 bisectors  of  a  triangle  meet.  It  is 
 generally  denoted  by  ‘I’.  From  the 
 in-center  the  in-circle  of  a  triangle  is 
 drawn.  The  radius  of  the  in-circle  is  equal 
 to ID = IE = IF. 
 Important Results: 
 ?  +  (for  Internal  ?  ?????? = 90° 
 ?  ?? 
 2 
 angle bisectors) 
 ?  (for  ?  ?????? = 90° -   
 ?  ?? 
 2 
 external angle bisectors) 
 ?  = 
 ???? 
 ???? 
 ???? 
 ???? 
 ?  In-radius r  = 
 ????????     ????     ???????????????? 
 ???????? - ??????????????????     ????     ???????????????? 
 ?  AI : ID = AB + AC : BC 
 BI : IE = AB + BC : AC 
 CI : IF = AC + BC : AB 
 Length of angle bisector(AD)    •
 = 
 2     ×     ????        ×     ????     ?????? ?
 ????       +       ???? 
 Circum-center :- 
 The  point  at  which  the  perpendicular 
 bisectors  of  the  sides  of  the  triangle 
 meet. 
 ?  From  the  circum-center  the 
 circumcircle  of  a  triangle  is 
 drawn.  The  radius  of  the 
 circum-circle  is  equal  to  PG  = 
 RG = QG. 
 ?  Since  the  angle  formed  at  the 
 center  of  a  circle  is  double  of 
 the  angle  formed  at  the 
 circumference, we have:: 
 ,  ?  ?????? = 2?  ??  ?  ?????? = 2?  ?? 
 and  .     ?  ?????? = 2?  ?? 
 ?  For  an  obtuse  angle  triangle,  the 
 circumcenter  lies  outside  the 
 triangle. 
 Note:  A  line  which  is  perpendicular  to 
 another  line  and  also  bisects  it  into  two 
 equal  parts  is  called  a  perpendicular 
 bisector. 
 Centroid:  It  is  the  point  of  intersection  of 
 the  medians  of  a  triangle.  It  is  also  called 
 the center of mass. 
 ?  The  median  is  a  line  drawn  from 
 the  vertex  of  a  triangle  to  the 
 mid  –  point  of  the  opposite 
 side. 
 ?  Each  median  is  divided  in  the 
 ratio of 2 : 1 at the centroid. 
 ?  Median  divides  the  triangle  into 
 two equal areas. 
 ?  Apollonius theorem: 
 ? 
 ???? 
 2 
+ ???? 
 2 
= 2 ( ???? 
 2 
+ ???? 
 2 
)
 Other key points for medians: 
 ?  All  the  medians  of  a  triangle 
 divide  it  into  six  parts  with  equal 
 areas. 
 Therefore,  ????????     ????     ?  ?????? 
= 6  ×  ????????  ????  ??????  ????  ??h?? 
    ??????????????     ?????????? .
 ?  ????????     ????     ?  ?????? 
 = 
 1 
 12 
    ×     ????????     ????  ?  ?????? 
 ?  FE  divides  the  line  AG  in  the 
 ratio of 3 : 1 i.e.  . 
 ???? 
 ???? 
=
 3 
 1 
 ?  Area  of  ABG  =  Area  of  BCG  = ? ?
 Area of  ACG ?
 ?  Area of  DEF =  ABC ?
 1 
 4 
?
 Note  :  In  case  of  Equilateral  triangle, 
 orthocenter,  in-center,  circum-center  and 
 centroid lie at the same point. 
 Other Facts about triangles: 
 1).  Sum of two sides is always greater 
 than the third side. 
 E.g.  AB  +  BC  >  CA,  AB  +  AC  >  BC,  BC  +  CA 
 > AB. 
 2).  Difference of two sides is always less 
 than the third side. 
 E.g.  AB  -  BC  <  CA,  AB  -  AC  <  BC,  BC  -  CA 
 < AB. 
 3).  In  ABC,  B  >  C,  if  AM  is  the ? ?  ? 
 bisector of  BAC and AN  BC, then  ? ?
 MAN =  (  B -  C)  ? 
 1 
 2 
 ?  ? 
 4).  In  right  angled  ABC,  where  BD  is  an ?
 altitude  to  hypotenuse  AC,  then  Altitude 
 BD =  . 
 ????    ×    ???? 
 ???? 
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 Similarly, if BE is a median drawn to 
 hypotenuse AC, then BE =  AC. (Length 
 1 
 2 
 of  median  is  half  the  length  of 
 hypotenuse) 
 5).  In  right  angled  ABC,  AD  is ?
 perpendicular to BC, then 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 Formulas  of triangle :- 
 In case of equilateral triangle, 
 Area(A) =  (side) 
 2 
•
 3 
 4 
 Inradius(r) = •
 ????????    
 2  3 
 circumradius(R) = •
 ????????    
 3 
 height(h) =  × side •
 3 
 2 
 I  n case of isosceles triangle, 
 Area(A) = •
 ???? 
 4 
 4  ???? 
 2 
- ???? 
 2 
 Where AB = AC 
 Inradius(r) = •
 ????????     ????     ????????????????    
 ???????? - ?????????????????? 
 circumradius(R) = •
 ????     ×     ????     ×     ???? 
 4     ×     ????????     ????     ????????????????    
 height(h) = • ???? 
 2 
-
 ???? 
 2 
 4 
 In case of scalene triangle, 
 Area(A) = • ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 Where a,b,c are sides of triangle 
 s = 
 ??    +    ??    +    ?? 
 2 
 In case of right angle triangle , 
 Area(A) =  × Base × height •
 1 
 2 
 Inradius(r) = •
 ??????????????????????????    +    ????????    -    h?????????????????? 
 2 
 circumradius(R) = •
 h?????????????????? 
 2 
 Note :-  Distance between circumcentre 
 and incentre of any triangle  = 
 ?? 
 2 
   -    2  ???? 
 Sine rule :- 
 =  =  =  2R 
 ?? 
 ???????? 
 ?? 
 ???????? 
 ?? 
 ???????? 
 Where R =  circumradius of triangle 
 Cosine rule :- 
 CosA = 
 ?? 
 2 
+ ?? 
 2 
-    ?? 
 2 
 2  ???? 
 CosB = 
 ?? 
 2 
+    ?? 
 2 
-    ?? 
 2 
 2  ???? 
 CosC = 
 ?? 
 2 
   +    ?? 
 2 
-    ?? 
 2 
 2  ???? 
 If  P  is  any  point  inside  the  equilateral •
 triangle  ,  the  sum  of  altitude  from  the 
 point  P  to  the  sides  AB,  BC  and  AC  equal 
 to the sides of the triangle. 
 +  +  =  a  h 
 1 
 h 
 2 
 h 
 3 
 3 
 2 
 Where a = side of equilateral triangle 
 if  ABC is right angle triangle and AD, • ?
 CE are medians then: 
 (i)  4(AD 
 2 
 + CE 
 2 
 ) = 5 AC 
 2 
 (ii)  AD 
 2 
 + CE 
 2 
 = AC 
 2 
 + ED 
 2 
 If BE and CD are medians and •
 perpendicular to each other then: 
 AB 
 2 
 + AC 
 2 
 = 5 BC 
 2 
 If length of median of triangles are •
 given then area of triangle =  × area of 
 4 
 3 
 triangle formed by medians. 
 CIRCLE 
 Properties of circles :- 
 1).  Angles  in  the  same  segment  of  a 
 circle  from  the  same  base  are  always 
 equal. 
 2).  Angle  dropped  on  the  circumference 
 of  a  circle  with  the  diameter  as  base  is 
 always a right angle. 
 3).  Two  chords  AB  and  CD  of  a  circle 
 intersect,  internally  or  externally,  at  E 
 then: 
 ????  ×  ???? = ????  ×  ???? 
 4).  The  tangent  to  a  circle  at  a  point  on 
 its  circumference  is  perpendicular  to  the 
 radius at that point. 
 OI  PT ?
 5).  Two  tangents  PA  and  PB  are  drawn 
 from  an  external  point  P  on  a  circle  with 
 center O is equal.  PA = PB 
 6).  For  a  tangent,  PT,  and  a  secant,  PB, 
 drawn  to  a  circle  from  a  point  P ,  we  have. 
 ???? 
 2 
= ????  ×  ???? 
 7).  Angle  made  by  a  chord  with  a  tangent 
 is  always  equal  to  the  angle  dropped  on 
 any  point  of  circumference  taking  the 
 chord as base. 
 ?  ?????? = ?  ?????? 
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 8).  In the following case: 
    ????     ×     ???? + ????  ×  ???? = ???? 
 2 
 9).  If  two  circles  touch  each  other 
 externally,  then  distance  between  their 
 centres is the sum of their radius. 
 Here, PQ = R + r 
 10).  If  two  circles  touch  each  other 
 internally,  then  distance  between  their 
 centres is the difference of their radius. 
 Here, PQ = R - r 
 Direct common Tangents: 
 Length  of  direct  common  tangent  (L)  = 
 ?? 
 2 
- ( ?? 
 1 
- ?? 
 2 
)
 2 
 Where,  d  =  distance  between  two  centers 
 and  are radii of the circles.  ?? 
 1 
,       ?? 
 2 
 Transverse common Tangents:- 
 Length  of  transverse  tangent  (L)  = 
 ?? 
 2 
- ( ?? 
 1 
+ ?? 
 2 
)
 2 
 Cyclic Quadrilateral  :- 
 A quadrilateral drawn inside a circle with 
 its vertices lying on the circumference. 
 Sum of opposite angles = 180  ° 
 i.e.  ?  ?? + ?  ?? = ?  ?? + ?  ?? = 180° 
 Important  result:  In  the  following  type  of 
 cyclic  quadrilateral  remember  the 
 relationship  between  the  angles,  as 
 shown: 
 QUADRILATERAL 
 Any  closed  ?gure  that  has  four  sides  is 
 called  a  quadrilateral.  Depending  on 
 length  of  sides,  orientation,  they  can  be 
 classi?ed as followed: 
 1).  Parallelogram  :  Opposite  sides  are 
 equal and parallel. 
 ?  AB = DC and AD = BC 
 ?  ?  ?? + ?  ?? = ?  ?? + ?  ?? =
 ?  ?? + ?  ?? = ?  ?? + ?  ?? = 180° 
 ?  and  ?  ?? = ?  ??  ?  ?? = ?  ?? 
 ?  The  diagonals  bisect  each  other  i.e. 
 AO = OC and OB = OD 
 ?  Area  of  parallelogram  =  Base ×
 Altitude 
 ?  Sum  of  squares  of  all  four  sides  = 
 Sum of squares of diagonals, So, 
 +  =  +  +  +  ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ?  Square,  Rectangle  and  Rhombus  are 
 parallelogram. 
 ?  All  rectangles  are  parallelogram  but 
 all parallelogram are not rectangles. 
 2).  Trapezium :  Only one pair of opposite 
 sides are parallel. 
 Area of trapezium ABCD 
 = 
 1 
 2 
    ×     ??????     ????     ????????????????     ????????     ×     h??????h?? 
 = 
 1 
 2 
    ×    ( ???? + ???? )    ×     h 
 ?  If  E  is  the  midpoint  of  and  F  is  ???? 
 the midpoint of  , then  ???? 
 EF =  . 
 ????    +    ???? 
 2 
 ?  If  P  is  the  midpoint  of  diagonal  ???? 
 and  Q  is  the  midpoint  of  diagonal  ???? 
 , then 
 PQ = 
 ????    -    ???? 
 2 
 3).  Rhombus  :  All  sides  are  equal  and 
 opposite sides are parallel to each other. 
 ?  ?? + ?  ?? = ?  ?? + ?  ?? =
 ?  ?? + ?  ?? = ?  ?? + ?  ?? = 180° 
 ?  and  ?  ?? = ?  ??  ?  ?? = ?  ?? 
 ?  4  ?? 
 2 
= ?? 
 1 
 2 
+ ?? 
 2 
 2 
 i.e.  Sum  of  squares  of  sides  =  Sum  of 
 squares of diagonals. 
 ?  Diagonals  bisect  each  other  at  right 
 angles  and  form  four  right  angled 
 triangles with equal areas. 
 ????????     ????     ?  ?????? =    ?  ?????? = ?  ?????? 
 = = ?  ?????? 
 1 
 4 
 ×  ????????     ????     ??h??????????     ???????? 
 ?  The  diagonals  of  Rhombus  are  not 
 of equal length. 
 4).  Square:  All  sides  are  equal  in  length 
 and  adjacent  sides  are  perpendicular  to 
 each other. 
 ?  AB = BC = CD = AD and 
 ?  ?? = ?  ?? = ?  ?? = ?  ?? = 90° 
 ?  Diagonals  bisect  each  other  at  right 
 angles  and  form  four  right  angled 
 isosceles triangles. 
 ?  Diagonals are of equal length: 
 i.e. AC = BD. 
 ?  In  a  rectangle  the  diagonals  do  not 
 intersect at right angles. 
 5).  Polygons: 
 ?  Convex  polygon:  All  its  interior 
 angles are less than 180  .  ° 
 ?  Concave  polygon:  At  least  one 
 angle is more than 180  .  ° 
 ?  Regular  Polygons:  All  sides  and 
 angles are equal. 
 * Sum of all exterior angles =  .  360° 
 * Each exterior angle =  ,  where n = 
 360° 
 ?? 
 number of sides. 
 * Exterior angle + Interior angle = 180  ° 
 * In case of convex polygon, sum of all 
 interior angles = ( 2  ?? - 4 )    ×     90° 
 * Number of diagonals  = 
 ?? ( ?? - 3 )
 2 
 Where ,   n = number of sides. 
 ALTERNATE SEGMENT 
 THEOREM 
 The Alternate Segment Theorem states 
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 Pinnacle  Day: 30th - 38th  Geometry 
 Geometry 
 LINES 
 1).  Parallel  Lines  :  Two  or  more  lines 
 which  will  never  meet  just  like  railway 
 track. 
 2).  Transversal  Lines:  A  line  which  cuts 
 parallel  lines  as  shown  in  the  ?gure 
 below. 
 ?  EF  is  the  transverse  line.  AB  and 
 CD  are  parallel  lines.  Symbol  of 
 parallel  lines  is  ‘||’.  To  denote  a 
 line  symbols  like  or  simply  ???? 
 AB are used. 
 ?  Let’s  take  a  look  at  various 
 angles  made  by  the  transverse 
 line and the parallel lines : 
 (i)  Corresponding  Angles:-  and  ,  ?1  ?5  ?2 
 and  ,  and  ,  and  are  pairs  ?6  ?4  ?8  ?3  ?7 
 of corresponding angles. 
 Corresponding  angles  are  equal.  E.g. •
 ?1 = ?5 
 (ii)  Alternate  Angles:-  E.g.  (  ),  (  ?1 = ?7 
 ), (  ),  (  ).  ?2 = ?8  ?4 = ?6  ?3 = ?5 
 (iii)  Vertically opposite angles  :- 
 E.g. (  ), (  ), (  ),  ?1 = ?3  ?2 = ?4  ?6 = ?8 
 (  )  ?5 = ?7 
 (iv)  Adjacent  angles:-  in  ?1 + ?2 = 180° 
 this  case  as  these  are  linear  pairs.  It  is  not 
 necessary that their sum should be  .  180° 
 (v)  Sum  of  Interior  angles  on  same  side  = 
 2×  right  angles  =  .  The  angle,  made  180° 
 by bisectors of interior angle will be  .  90° 
 (vi)  Sum  of  Exterior  angles  on  same  side 
 =  180° 
 Note:  From  a  point,  an  in?nite  number  of 
 lines can be drawn. 
 3).  Internal  angle  bisector  and  External 
 angle  bisector  .  In  the  given  ?gure  AB  is 
 the  internal  angle  bisector  and  AC  is  the 
 external angle bisector. 
 ANGLES 
 Various types of angles are as follows: 
 1).  Acute Angle: Angles which are less 
 than  .  90° 
 2).  Right Angle:  angle is called right  90° 
 angle. 
 3).  Obtuse Angle: Angles greater than 
 and less than  .  90°  180° 
 4).  Straight  Angles:  Angle  equal  to  .  180° 
 5).  Re?ex  angle:  Angles  greater  than 
 and less than  .  180°  360° 
 6).  Complete Angle:  angle is called  360° 
 Complete angle or whole angle. 
 7).  Complementary  Angles:  Sum  of  two 
 angles is equal to  90° .
 8).  Supplementary  Angles:  Sum  of  the 
 two angles is equal to  180° .
 TYPES OF TRIANGLES 
 1). Based on sides:- 
 ?  Equilateral  Triangles:  All  three 
 sides are equal to each other. 
 ?  Isosceles  Triangles  :  Any  two 
 sides are equal to each other. 
 ?  Scalene  Triangle:  All  three  sides 
 are different in length. 
 2). Based on angles :- 
 ?  Right  angle  triangle  :-  One  of  the  angle 
 is  90° .
 ?  Obtuse  angled  triangle  :-  One  angle  is 
 more than  90° .
 ?  Acute  angled  triangle  :-  All  three 
 angles are less than  90° .
 Properties of Triangles 
 [1.]  If  a  line  DE  intersects  two  sides  of  a 
 triangle  AB  and  AC  at  D  and  E 
 respectively,  then  :  =  and 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 ???? 
 =  .  Also  if  D  and  E  are  mid  points  of 
 ???? 
 ???? 
 AB and AC respectively, then: 
 =  and DE will be parallel to BC.  ???? 
 ???? 
 2 
 E.g.  Let  AD  =  2.5  units;  DB  =  5,  AE  =  2  and 
 BC = 9, then EC = ? and DE = ? 
 =  = 
 ???? 
 ???? 
 ???? 
 ???? 
? ???? = ????     ×    
 ???? 
 ???? 
 2     ×    
 5 
 2 . 5 
 =  Similarly DE = 3 units.  4     ?????????? .
 [2.]  In  case  of  Internal  angle  bisector: 
 or, 
 ???? 
 ???? 
=
 ???? 
 ???? 
 ???? 
 ???? 
=
 ???? 
 ???? 
 Here  PS  is  the  internal  angle  bisector  and 
 a  common  side  of  the  triangles 
 .  ?     ??????     ??????     ?  ?????? 
 [3.]  In case of  Exterior angle bisector : 
 = 
 ???? 
 ???? 
 ???? 
 ???? 
 PS is the external angle bisector. 
 Congruency of Triangles 
 Two triangles will be congruent if: 
 1.  SSS:  (Side  –  Side  –  Side  rule):  When 
 all three sides are equal. 
 2.  SAS:  (Side  –  Angle  –  Side  rule):  When 
 two  sides  and  one  angle  are  equal. 
 3.  ASA:  (Angle  –  Side  –  Angle  rule): 
 When two angles and one side are equal. 
 4.  RHS:  (Right  angle  –  Hypotenuse  – 
 Side  rule):  When  one  side  and 
 hypotenuse  of  the  right  angled  triangle 
 are equal. 
 Similarity of Triangles 
 1).  Similar  triangles  are  triangles  that 
 have the same shape, but their sizes may 
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 vary. 
 2).  In  congruency  the  triangles  are  mirror 
 images  of  each  other.  We  can  say  that  all 
 congruent  triangles  are  similar  but  the 
 vice versa is not true. 
 3).  Important  properties  of  similar 
 triangles: 
 ?  = 
 ????????     ????     ?  ?????? 
 ????????     ????     ?  ?????? 
 ???? 
 2 
 ???? 
 2 
 = 
 ???? 
 2 
 ???? 
 2 
   =   
 ???? 
 2 
 ???? 
 2 
 ?  = 
 ??????????????????     ????     ?  ?????? 
 ??????????????????     ????     ?  ?????? 
 ???? 
 ???? 
=
 ???? 
 ???? 
   =   
 ???? 
 ???? 
 Altitude : 
 ?  It  is  also  known  as  height.  The 
 line  segment  drawn  from  the 
 vertex  of  a  triangle 
 perpendicular  to  its  opposite 
 side  is  called  an  Altitude  of  a 
 triangle. 
 Orthocenter: 
 ?  The  point  at  which  the  altitudes 
 of  a  triangle  intersect  is  called 
 an Orthocenter. 
 ?  Generally  Orthocenter  is 
 denoted by ‘O’. 
 ?  Here,  AOC = 180° -  B  ?  ? 
 ?  In  the  case  of  a  right  angled 
 triangle,  the  orthocenter  lies  at 
 the vertex of  right angle. 
 ?  In  case  of  an  obtuse  angled 
 triangle,  the  orthocenter  lies 
 outside the triangle. 
 In-center: 
 The  point  at  which  the  internal  angle 
 bisectors  of  a  triangle  meet.  It  is 
 generally  denoted  by  ‘I’.  From  the 
 in-center  the  in-circle  of  a  triangle  is 
 drawn.  The  radius  of  the  in-circle  is  equal 
 to ID = IE = IF. 
 Important Results: 
 ?  +  (for  Internal  ?  ?????? = 90° 
 ?  ?? 
 2 
 angle bisectors) 
 ?  (for  ?  ?????? = 90° -   
 ?  ?? 
 2 
 external angle bisectors) 
 ?  = 
 ???? 
 ???? 
 ???? 
 ???? 
 ?  In-radius r  = 
 ????????     ????     ???????????????? 
 ???????? - ??????????????????     ????     ???????????????? 
 ?  AI : ID = AB + AC : BC 
 BI : IE = AB + BC : AC 
 CI : IF = AC + BC : AB 
 Length of angle bisector(AD)    •
 = 
 2     ×     ????        ×     ????     ?????? ?
 ????       +       ???? 
 Circum-center :- 
 The  point  at  which  the  perpendicular 
 bisectors  of  the  sides  of  the  triangle 
 meet. 
 ?  From  the  circum-center  the 
 circumcircle  of  a  triangle  is 
 drawn.  The  radius  of  the 
 circum-circle  is  equal  to  PG  = 
 RG = QG. 
 ?  Since  the  angle  formed  at  the 
 center  of  a  circle  is  double  of 
 the  angle  formed  at  the 
 circumference, we have:: 
 ,  ?  ?????? = 2?  ??  ?  ?????? = 2?  ?? 
 and  .     ?  ?????? = 2?  ?? 
 ?  For  an  obtuse  angle  triangle,  the 
 circumcenter  lies  outside  the 
 triangle. 
 Note:  A  line  which  is  perpendicular  to 
 another  line  and  also  bisects  it  into  two 
 equal  parts  is  called  a  perpendicular 
 bisector. 
 Centroid:  It  is  the  point  of  intersection  of 
 the  medians  of  a  triangle.  It  is  also  called 
 the center of mass. 
 ?  The  median  is  a  line  drawn  from 
 the  vertex  of  a  triangle  to  the 
 mid  –  point  of  the  opposite 
 side. 
 ?  Each  median  is  divided  in  the 
 ratio of 2 : 1 at the centroid. 
 ?  Median  divides  the  triangle  into 
 two equal areas. 
 ?  Apollonius theorem: 
 ? 
 ???? 
 2 
+ ???? 
 2 
= 2 ( ???? 
 2 
+ ???? 
 2 
)
 Other key points for medians: 
 ?  All  the  medians  of  a  triangle 
 divide  it  into  six  parts  with  equal 
 areas. 
 Therefore,  ????????     ????     ?  ?????? 
= 6  ×  ????????  ????  ??????  ????  ??h?? 
    ??????????????     ?????????? .
 ?  ????????     ????     ?  ?????? 
 = 
 1 
 12 
    ×     ????????     ????  ?  ?????? 
 ?  FE  divides  the  line  AG  in  the 
 ratio of 3 : 1 i.e.  . 
 ???? 
 ???? 
=
 3 
 1 
 ?  Area  of  ABG  =  Area  of  BCG  = ? ?
 Area of  ACG ?
 ?  Area of  DEF =  ABC ?
 1 
 4 
?
 Note  :  In  case  of  Equilateral  triangle, 
 orthocenter,  in-center,  circum-center  and 
 centroid lie at the same point. 
 Other Facts about triangles: 
 1).  Sum of two sides is always greater 
 than the third side. 
 E.g.  AB  +  BC  >  CA,  AB  +  AC  >  BC,  BC  +  CA 
 > AB. 
 2).  Difference of two sides is always less 
 than the third side. 
 E.g.  AB  -  BC  <  CA,  AB  -  AC  <  BC,  BC  -  CA 
 < AB. 
 3).  In  ABC,  B  >  C,  if  AM  is  the ? ?  ? 
 bisector of  BAC and AN  BC, then  ? ?
 MAN =  (  B -  C)  ? 
 1 
 2 
 ?  ? 
 4).  In  right  angled  ABC,  where  BD  is  an ?
 altitude  to  hypotenuse  AC,  then  Altitude 
 BD =  . 
 ????    ×    ???? 
 ???? 
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 Similarly, if BE is a median drawn to 
 hypotenuse AC, then BE =  AC. (Length 
 1 
 2 
 of  median  is  half  the  length  of 
 hypotenuse) 
 5).  In  right  angled  ABC,  AD  is ?
 perpendicular to BC, then 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 ?  =  ??  ?? 
 2 
 ???? × ???? 
 Formulas  of triangle :- 
 In case of equilateral triangle, 
 Area(A) =  (side) 
 2 
•
 3 
 4 
 Inradius(r) = •
 ????????    
 2  3 
 circumradius(R) = •
 ????????    
 3 
 height(h) =  × side •
 3 
 2 
 I  n case of isosceles triangle, 
 Area(A) = •
 ???? 
 4 
 4  ???? 
 2 
- ???? 
 2 
 Where AB = AC 
 Inradius(r) = •
 ????????     ????     ????????????????    
 ???????? - ?????????????????? 
 circumradius(R) = •
 ????     ×     ????     ×     ???? 
 4     ×     ????????     ????     ????????????????    
 height(h) = • ???? 
 2 
-
 ???? 
 2 
 4 
 In case of scalene triangle, 
 Area(A) = • ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 Where a,b,c are sides of triangle 
 s = 
 ??    +    ??    +    ?? 
 2 
 In case of right angle triangle , 
 Area(A) =  × Base × height •
 1 
 2 
 Inradius(r) = •
 ??????????????????????????    +    ????????    -    h?????????????????? 
 2 
 circumradius(R) = •
 h?????????????????? 
 2 
 Note :-  Distance between circumcentre 
 and incentre of any triangle  = 
 ?? 
 2 
   -    2  ???? 
 Sine rule :- 
 =  =  =  2R 
 ?? 
 ???????? 
 ?? 
 ???????? 
 ?? 
 ???????? 
 Where R =  circumradius of triangle 
 Cosine rule :- 
 CosA = 
 ?? 
 2 
+ ?? 
 2 
-    ?? 
 2 
 2  ???? 
 CosB = 
 ?? 
 2 
+    ?? 
 2 
-    ?? 
 2 
 2  ???? 
 CosC = 
 ?? 
 2 
   +    ?? 
 2 
-    ?? 
 2 
 2  ???? 
 If  P  is  any  point  inside  the  equilateral •
 triangle  ,  the  sum  of  altitude  from  the 
 point  P  to  the  sides  AB,  BC  and  AC  equal 
 to the sides of the triangle. 
 +  +  =  a  h 
 1 
 h 
 2 
 h 
 3 
 3 
 2 
 Where a = side of equilateral triangle 
 if  ABC is right angle triangle and AD, • ?
 CE are medians then: 
 (i)  4(AD 
 2 
 + CE 
 2 
 ) = 5 AC 
 2 
 (ii)  AD 
 2 
 + CE 
 2 
 = AC 
 2 
 + ED 
 2 
 If BE and CD are medians and •
 perpendicular to each other then: 
 AB 
 2 
 + AC 
 2 
 = 5 BC 
 2 
 If length of median of triangles are •
 given then area of triangle =  × area of 
 4 
 3 
 triangle formed by medians. 
 CIRCLE 
 Properties of circles :- 
 1).  Angles  in  the  same  segment  of  a 
 circle  from  the  same  base  are  always 
 equal. 
 2).  Angle  dropped  on  the  circumference 
 of  a  circle  with  the  diameter  as  base  is 
 always a right angle. 
 3).  Two  chords  AB  and  CD  of  a  circle 
 intersect,  internally  or  externally,  at  E 
 then: 
 ????  ×  ???? = ????  ×  ???? 
 4).  The  tangent  to  a  circle  at  a  point  on 
 its  circumference  is  perpendicular  to  the 
 radius at that point. 
 OI  PT ?
 5).  Two  tangents  PA  and  PB  are  drawn 
 from  an  external  point  P  on  a  circle  with 
 center O is equal.  PA = PB 
 6).  For  a  tangent,  PT,  and  a  secant,  PB, 
 drawn  to  a  circle  from  a  point  P ,  we  have. 
 ???? 
 2 
= ????  ×  ???? 
 7).  Angle  made  by  a  chord  with  a  tangent 
 is  always  equal  to  the  angle  dropped  on 
 any  point  of  circumference  taking  the 
 chord as base. 
 ?  ?????? = ?  ?????? 
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 8).  In the following case: 
    ????     ×     ???? + ????  ×  ???? = ???? 
 2 
 9).  If  two  circles  touch  each  other 
 externally,  then  distance  between  their 
 centres is the sum of their radius. 
 Here, PQ = R + r 
 10).  If  two  circles  touch  each  other 
 internally,  then  distance  between  their 
 centres is the difference of their radius. 
 Here, PQ = R - r 
 Direct common Tangents: 
 Length  of  direct  common  tangent  (L)  = 
 ?? 
 2 
- ( ?? 
 1 
- ?? 
 2 
)
 2 
 Where,  d  =  distance  between  two  centers 
 and  are radii of the circles.  ?? 
 1 
,       ?? 
 2 
 Transverse common Tangents:- 
 Length  of  transverse  tangent  (L)  = 
 ?? 
 2 
- ( ?? 
 1 
+ ?? 
 2 
)
 2 
 Cyclic Quadrilateral  :- 
 A quadrilateral drawn inside a circle with 
 its vertices lying on the circumference. 
 Sum of opposite angles = 180  ° 
 i.e.  ?  ?? + ?  ?? = ?  ?? + ?  ?? = 180° 
 Important  result:  In  the  following  type  of 
 cyclic  quadrilateral  remember  the 
 relationship  between  the  angles,  as 
 shown: 
 QUADRILATERAL 
 Any  closed  ?gure  that  has  four  sides  is 
 called  a  quadrilateral.  Depending  on 
 length  of  sides,  orientation,  they  can  be 
 classi?ed as followed: 
 1).  Parallelogram  :  Opposite  sides  are 
 equal and parallel. 
 ?  AB = DC and AD = BC 
 ?  ?  ?? + ?  ?? = ?  ?? + ?  ?? =
 ?  ?? + ?  ?? = ?  ?? + ?  ?? = 180° 
 ?  and  ?  ?? = ?  ??  ?  ?? = ?  ?? 
 ?  The  diagonals  bisect  each  other  i.e. 
 AO = OC and OB = OD 
 ?  Area  of  parallelogram  =  Base ×
 Altitude 
 ?  Sum  of  squares  of  all  four  sides  = 
 Sum of squares of diagonals, So, 
 +  =  +  +  +  ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ??  ?? 
 2 
 ?  Square,  Rectangle  and  Rhombus  are 
 parallelogram. 
 ?  All  rectangles  are  parallelogram  but 
 all parallelogram are not rectangles. 
 2).  Trapezium :  Only one pair of opposite 
 sides are parallel. 
 Area of trapezium ABCD 
 = 
 1 
 2 
    ×     ??????     ????     ????????????????     ????????     ×     h??????h?? 
 = 
 1 
 2 
    ×    ( ???? + ???? )    ×     h 
 ?  If  E  is  the  midpoint  of  and  F  is  ???? 
 the midpoint of  , then  ???? 
 EF =  . 
 ????    +    ???? 
 2 
 ?  If  P  is  the  midpoint  of  diagonal  ???? 
 and  Q  is  the  midpoint  of  diagonal  ???? 
 , then 
 PQ = 
 ????    -    ???? 
 2 
 3).  Rhombus  :  All  sides  are  equal  and 
 opposite sides are parallel to each other. 
 ?  ?? + ?  ?? = ?  ?? + ?  ?? =
 ?  ?? + ?  ?? = ?  ?? + ?  ?? = 180° 
 ?  and  ?  ?? = ?  ??  ?  ?? = ?  ?? 
 ?  4  ?? 
 2 
= ?? 
 1 
 2 
+ ?? 
 2 
 2 
 i.e.  Sum  of  squares  of  sides  =  Sum  of 
 squares of diagonals. 
 ?  Diagonals  bisect  each  other  at  right 
 angles  and  form  four  right  angled 
 triangles with equal areas. 
 ????????     ????     ?  ?????? =    ?  ?????? = ?  ?????? 
 = = ?  ?????? 
 1 
 4 
 ×  ????????     ????     ??h??????????     ???????? 
 ?  The  diagonals  of  Rhombus  are  not 
 of equal length. 
 4).  Square:  All  sides  are  equal  in  length 
 and  adjacent  sides  are  perpendicular  to 
 each other. 
 ?  AB = BC = CD = AD and 
 ?  ?? = ?  ?? = ?  ?? = ?  ?? = 90° 
 ?  Diagonals  bisect  each  other  at  right 
 angles  and  form  four  right  angled 
 isosceles triangles. 
 ?  Diagonals are of equal length: 
 i.e. AC = BD. 
 ?  In  a  rectangle  the  diagonals  do  not 
 intersect at right angles. 
 5).  Polygons: 
 ?  Convex  polygon:  All  its  interior 
 angles are less than 180  .  ° 
 ?  Concave  polygon:  At  least  one 
 angle is more than 180  .  ° 
 ?  Regular  Polygons:  All  sides  and 
 angles are equal. 
 * Sum of all exterior angles =  .  360° 
 * Each exterior angle =  ,  where n = 
 360° 
 ?? 
 number of sides. 
 * Exterior angle + Interior angle = 180  ° 
 * In case of convex polygon, sum of all 
 interior angles = ( 2  ?? - 4 )    ×     90° 
 * Number of diagonals  = 
 ?? ( ?? - 3 )
 2 
 Where ,   n = number of sides. 
 ALTERNATE SEGMENT 
 THEOREM 
 The Alternate Segment Theorem states 
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 that  :  An  angle  between  a  tangent  and  a 
 chord  through  a  point  of  contact  is  equal 
 to the angle in the alternate segment. 
 Here,  in  above  diagram,  is  tangent.  ???? 
 and  are  chord  which  touches  ????     ????    
 tangent  at  point  P .  Hence,  TPQ  =  ????  ?  ? 
 PRQ and  CPR =  PQR  ?  ? 
 Important formulas :- 
 Circle :- 
 If  r  be  the  radius  and  O  be  the  center  of •
 the circle. 
 Diameter = 2 × radius 
 Area  =  pr 
 2 
 Circumference = 2pr 
 Semi - Circle :- 
 Diameter = 2 × radius 
 Perimeter =  + 2)r ( p 
 Area = 
 p  ?? 
 2 
 2 
 Sector :- 
 If ?AOB = ?
 Perimeter  =  [2r +  (2 ?? r)] 
?
 360 
×
 Area =  × pr 
 2 
?
 360° 
 Two  Chords  AB  and  CD  of  a  circle  with •
 center  O,  intersect  each  other  at  P .  If 
 ?AOC  =  x°  and  ?BOD  =  y°  then  the  value 
 of  ?BPD is :- 
 ?BPD = 
 ??    +    ?? 
 2 
 Chords  are  intersecting  at  external •
 points. 
 ?BPD = 
 ??    -    ?? 
 2 
 Variety Questions 
 Q.1.  If  a  transversal  intersects  two 
 parallel  lines,  and  the  difference  between 
 two  interior  angles  on  the  same  side  of 
 the  transversal  is  40°,  then  ?nd  the 
 smallest angle among the interior angles. 
 SSC CHSL Tier II  (10/01/2024) 
 (a) 70°  (b) 50°  (c) 40°  (d) 60° 
 Q.2.  Which  of  the  following  statements  is 
 NOT true ? 
 SSC CHSL Tier II  (02/11/2023) 
 (a) If the sides of a triangle are in the 
 ratio 13 : 5 : 12, then two of its angles 
 are acute angles. 
 (b) The triangle formed by connecting 
 the midpoints of the sides of a 
 triangle has one-fourth area of the 
 bigger triangle. 
 (c) The area of the triangle with the 
 largest side  always has an area less  ?? 
 than  . 
 ?? 
 2 
 2 
 (d) The area of an isosceles triangle with 
 sides 2a, 2a and 3a is  . 
 3  3  ?? 
 2 
 2 
 Q.3.  ?PQR  is  right  angled  at  Q  such  that 
 PQ  =  (x  -  y),  QR  =  x  and  PR  =  (x  +  y).  S  is  a 
 point  on  QR  such  that  QS  =  PQ  .  The  ratio 
 QS : SR for any values of x and y is: 
 SSC CPO 05/10/2023 (3rd Shift) 
 (a) 1 : 3  (b) 1 : 2  (c) 2 : 1  (d) 3 : 1 
 Q.4.  Two  circles  with  centres  P  and  Q  of 
 radii  6  cm  and  4  cm,  respectively,  touch 
 each  other  internally.  If  the  perpendicular 
 bisector  of  PQ  meets  the  bigger  circle  in 
 A and B, then the value of AB is: 
 SSC CPO 05/10/2023 (2nd Shift) 
 (a)  cm  (b)  cm  35  5 
 (c)  2  cm  (d) 2  cm  5  35 
 Q.5.  Two  circles  of  radius  13  cm  and  15 
 cm  intersect  each  other  at  points  A  and 
 B.  If  the  length  of  the  common  chord  is 
 12  cm,  then  what  is  the  distance  between 
 their centers? 
 SSC CPO 03/10/2023 (1st Shift) 
 (a)  (b)  145 + 184  131 + 181 
 (c)  (d)  145 + 169  133 + 189 
 Q.6.  M  is  the  mid-point  of  side  QR  of  a 
 parallelogram  PQRS  (P  being  on  the  top 
 left  hand,  followed  by  other  points  going 
 clockwise).  The  line  SM  is  drawn 
 intersecting  PQ  produced  at  T.  What  is 
 the  length  (in  terms  of  the  length  of  SR) 
 of PT? 
 SSC CHSL 10/08/2023 (1st Shift) 
 (a)  (b)  (c) 2SR     (d) SR 
 3  ???? 
 2 
 ???? 
 2 
 Q.7  .  Two  circles  with  diameters  50  cm 
 and  58  cm,  respectively,  intersect  each 
 other  at  points  A  and  B,  such  that  the 
 length  of  the  common  chord  is  40  cm. 
 Find  the  distance  (in  cm)  between  the 
 centres of these two circles. 
 SSC CHSL 08/08/2023 (3rd Shift) 
 (a) 34  (b) 37  (c) 36  (d) 35 
 Q.8.  PQ  is  a  chord  of  a  circle.  The 
 tangent  XR  at  X  on  the  circle  cuts  PQ 
 produced  at  R.  If  XR  =  12  cm,  PQ  =  x  cm, 
 QR = x - 2 cm, then x (in cm) is: 
 SSC CHSL 04/08/2023 (3rd Shift) 
 (a) 7  (b) 6  (c) 10  (d) 14 
 Q.9.  A  triangle  PQR  has  three  sides  equal 
 in  measurement  and  if  PM  is 
 perpendicular  to  QR,  then  which  of 
 following equality holds? 
 SSC CHSL 04/08/2023 (2nd Shift) 
 (a) 3PM 
 2 
 = 2PQ 
 2 
 (b) 3PQ 
 2 
 = 4PM 
 2 
 (c) 3PM 
 2 
 = 4PQ 
 2 
 (d) 3PQ 
 2 
 = 2PM 
 2 
 Q.10.  AB  =  8  cm  and  CD  =  6  cm  are  two 
 parallel  chords  on  the  same  side  of  the 
 centre  of  a  circle.  If  the  distance  between 
 them  is  2  cm,  then  the  radius  (in  cm)  of 
 the circle is : 
 SSC CHSL 02/08/2023 (4th Shift) 
 (a)  (b)  (c)  (d) 
 265 
 4 
 256 
 4 
 156 
 4 
 198 
 4 
 Q.11.  If  the  distance  between  the  centres 
 of  two  circles  is  12  cm  and  the  radii  are  5 
 cm  and  4  cm,  then  the  length  (in  cm)  of 
 the transverse common tangent is : 
 SSC CHSL 02/08/2023 (2nd Shift) 
 (a) 9   (b)  (c)  (d) 7  143  63 
 Q.12.  In  ,  the  straight  line  parallel  to ? ?????? 
 the  side  BC  meets  AB  and  AC  at  the 
 points  P  and  Q,  respectively.  If  AP  =  QC, 
 the  length  of  AB  is  16  cm  and  the  length 
 of  AQ  is  4  cm,  then  the  length  (in  cm)  CQ 
 is: 
 SSC CHSL 02/08/2023 (2nd Shift) 
 (a)  (b) (  )  2  21 + 2 
( )
 2     18 - 2 
 (c)  (d)  2  17 - 2 
( )
 2  19 + 2 
( )
 Q.13.  The  three  sides  of  a  triangle  are  9, 
 24  and  S  units.  Which  of  the  following  is 
 correct ? 
 SSC CHSL 02/08/2023 (1st Shift) 
 (a) 15 < S < 33  (b) 15  S < 33 =
 (c) 15 < S  33  (d) 15  S  33 = = =
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