prove that the chords of a circle can never be each others perpendicular bisector, unless they are diameters.

Asked by Joshi | 10th Nov, 2015, 09:50: AM

Expert Answer:

 
 
Let space AB space and space CD space be space the space two space perpendicular space bisecting space cords space of space the space circle.  space To space prove space that space AB space and space CD space are space diametre. space  space Join space points space straight A comma space straight D comma space straight B comma space straight C space to space form space straight a space cyclic space quadrilateral space ADBC.  In space increment ADP space and space increment BDP AP space equals space PB space space space space left parenthesis Given right parenthesis PD space equals space PD space space space left parenthesis Common space side right parenthesis angle BPD space equals space angle APD space space Therefore space increment ADP space approximately equal to space increment BDP space by space SAS space test So space angle PAD space equals angle PBD and space angle PDA space equals angle PDB  In space increment ACP space and space increment BCP AP space equals space PB space space space space left parenthesis Given right parenthesis PC space equals space PC space space space left parenthesis Common space side right parenthesis angle BPC space equals space angle APC space space Therefore space increment ACP space approximately equal to space increment BCP space by space SAS space test So space angle PAC space equals angle PBC and space angle PCA space equals angle PCB   Quadrilateral space ADBC space is space cyclic space quadrilateral. therefore angle CBD plus angle CAD equals 180 degree angle PBC plus angle PBD plus angle PAC plus angle PAD equals 180 degree angle PBC plus angle PBD plus angle PBC plus angle PBD equals 180 degree space space space space left parenthesis from space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis right parenthesis 2 angle PBC plus 2 angle PBD plus equals 180 degree space 2 angle CBD equals 180 degree angle CBD equals 90 degree space Hence space the space angle space made space by space the space chord space CD space space is space 90 degree Therefore space chord space CD space must space be space diameter.  Simillarly space we space can space prove space chord space AB space is space diameter.  Hence space proved.
 

Answered by Vijaykumar Wani | 10th Nov, 2015, 12:34: PM

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