OB is perpendicular bisector of line segment DE, FA perpendicular to OB and FE intersects OB at the point C. Prove that 1/OA + 1/OB= 2/ OC. (std 10 sum clearly explain and me mark you best brainlyest )
Answers
Follow this figure.
Step-by-step explanation:
Given: ▪︎ OB is perpendicular on DE and bisects it.
▪︎ FA is perpendicular on OB.
Therefore, BE = BD -{1} (Bisection by OB on DE)
First:
In △OAF and △OBD,
• ∠OAF = ∠OBD = 90° (Given)
• ∠BOD = ∠BOD (Common Angle)
Therefore, △OAF ~ △OBD (By AA Criterion of Similarity).
Therefore, OA/OB = AF/BD -{2} (Similarity ratio of Similar △s).
Second:
In △ACF and △BCE,
• ∠CAF = ∠CBE = 90° (Given)
• ∠ACF = ∠BCE (Vertically Opposite Angles)
Therefore, △ACF ~ △BCE (By AA Criterion of Similarity).
Therefore, CA/CB = AF/BE -{3} (Similarity ratio of Similar △s).
Third:
Following the equation {1}, we can equalize equations {2} and {3}.
OA = CA = AF = AF (BE=BD -{1})
OB CB BD BE
Therefore, OA = CA. -{4}.
OB CB
Fourth:
We can also write equation {4} in the manner :-
OA = OC - OA [=CA] (See the Figure).
OB OB - OC [=CB]
=> (OB - OC)OA = OC - OA(OB)
=> OB.OA - OC.OA = OB.OC - OB.OA
=> OB.OA + OB.OA = OB.OC + OC.OA
=> 2 OB.OA = OB.OC + OC.OA
Now, divide both L.H.S and R.H.S by OA.OB.OC
=> 2 OB.OA = OB.OC + OC.OA
OA.OB.OC OA.OB.OC OA.OB.OC
=> 2 = 1 + 1
OC OA OB
Hence, PROVED !! Yay...
Lovely question. Took me 2.5 hours to solve but I'm happy that I was able to help you [ and Myself too :-) ].
Hope it was helpful, do ask such questions, Makes me Happy. :-). Thank You. Cheers and Peace.
