In the given figure, OB is perpendicular bisector of the line segment DE,

FA⊥OB
and, FE intersects OB in C. Is the given statement true or false,

1

OA

+
1

OB

=
2

OC

.T

Question

In the given figure, OB is perpendicular bisector of the line segment DE,

FA⊥OB
   and, FE intersects OB in C. Is the given statement true or false,

1
    
     OA
   
   +
    1
    
     OB
   
   =
    2
    
     OC

.T

Accepted Answer

It is given that OB⊥DE, FA⊥OB and FE intersects OB in C.We shall first prove the similarities of ΔOAF, ΔODB, ΔAFC, and ΔBEC.Now considering, ΔOAF and ΔODB

∠OAF=∠OBD=
    
     90
    0

(OB⊥AF and OB⊥DE)∠FOA=∠DOB (common angle)Hence, ΔOAF≈ΔODB by RHS similarity criterion.Hence, we can write that, OAOB=AFDB=OFODSimilarly, for △AFC and △BEC∠FCA=∠BCE∠FAC=∠CBE=90⁰ So, with AA similarity, △AFC≅△BEC

So, 
    
     AF
    
     BE
   
   =
    
     AC
    
     CB
   
   =
    
     FC
    
     CE
   
   …(1)
  We know that DB=BE (Perpendicular bisector of DE is OB). So, from (1) AFDB=ACCB=FCCAlso we have,  OAOB=AFDB=OFOD

OA
    
     OB
   
   =
    
     AC
    
     CB
   
   =
    
     OC−OA
    
     OB−OC

→OA
    
     OB−OC
   =
    
     OC−OA
   OB

→OA.OB−OA.OC=OB.OC−OA.OB.
   ⇒2OAOB=OB.OC+OA.OCDividing by, OA.OB.OC 2OAOBOA.OB.OC=OA.OCOA.OB.OC+OB.OCOA.OB.OC ⇒1OA+1OB=2OC  Hence, from the given conditions it is derived that

1
    
     OA
   
   +
    1
    
     OB
   
   =
    2
    
     OC

.Therefore, the given statement is true i.e.,1OA+1OB=2OC.Hence, the correct option is 2.

In the given figure, OB is perpendicular bisector of the line segment DE, $F A ⊥ O B$ and, FE intersects OB in C. Is the given statement true or false, $1 O A + 1 O B = 2 O C$ .

T

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In the given figure, OB is perpendicular bisector of the line segment DE, FA⊥OB and, FE intersects OB in C. Is the given statement true or false, 1 OA + 1 OB = 2 OC .T