CBSE Class 12-science Answered

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is

Asked by vinitapahel123 | 09 Sep, 2015, 04:57: PM

Expert Answer

$C o n s i d e r space t h e space a b o v e space f i g u r e. L e t space triangle A B C space b e space a n space i s o s c e l e s space t r i a n g l e space w i t h space A B equals A C equals x space a n d space l e t space B C equals y L e t space apostrophe r apostrophe space b e space t h e space r a d i u s space o f space t h e space i n s c r i b e d space c i r c l e L e t space D comma space E space a n d space F space a r e space t h e space m i d p o i n t s space o f space t h e space s i d e s comma space A B comma space A C space a n d space B C space r e s p e c t i v e l y. S i n c e space A B C space i s space a n space i s o s c e l e s space t r i a n g l e comma space A F space b i s e c t s space B C rightwards double arrow B F equals C F equals y over 2 S i n c e space A F space i s space perpendicular B C comma space triangle A F C space i s space a space r i g h t space t r i a n g l e. B y space a p p l y i n g space P y t h a g o r a s space T h e o r e m comma space w e space h a v e comma A F squared plus C F squared equals A C squared rightwards double arrow open parentheses 3 r close parentheses squared plus open parentheses y over 2 close parentheses squared equals x squared... left parenthesis 1 right parenthesis F r o m space t r i a n g l e space A D O comma space w e space h a v e comma A D squared plus O D squared equals A O squared rightwards double arrow A D squared equals A O squared minus O D squared rightwards double arrow A D squared equals open parentheses 2 r close parentheses squared minus r squared rightwards double arrow A D squared equals 4 r squared minus r squared rightwards double arrow A D squared equals 3 r squared rightwards double arrow A D equals square root of 3 r L e n g t h s space o f space t h e space t a n g e n t s space f r o m space a n space e x t e r n a l space p o i n t space a r e space e q u a l. T h u s comma space B E equals B F space a n d space F C equals E C F r o m space t h e space f i g u r e comma space i t space i s space c l e a r space t h a t comma A D plus B E equals x rightwards double arrow square root of 3 r plus y over 2 equals x rightwards double arrow 2 square root of 3 r plus y equals 2 x rightwards double arrow y equals 2 x minus 2 square root of 3 r rightwards double arrow y equals 2 open parentheses x minus square root of 3 r close parentheses rightwards double arrow y over 2 equals open parentheses x minus square root of 3 r close parentheses.. left parenthesis 2 right parenthesis S u b s t i t u t i n g space t h e space v a l u e space y over 2 equals open parentheses x minus square root of 3 close parentheses space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma open parentheses 3 r close parentheses squared plus open parentheses x minus square root of 3 r close parentheses squared equals x squared rightwards double arrow 9 r squared plus x squared plus 3 r squared minus 2 square root of 3 x r equals x squared rightwards double arrow 9 r squared plus 3 r squared equals 2 square root of 3 x r rightwards double arrow 12 r squared equals 2 square root of 3 x r rightwards double arrow 6 r equals square root of 3 x rightwards double arrow x equals fraction numerator 6 r over denominator square root of 3 end fraction rightwards double arrow x equals 2 square root of 3 r S u b s t i t u t i n g space t h e space v a l u e space x equals 2 square root of 3 r space i n space e q u a t i o n space left parenthesis 2 right parenthesis comma space w e space h a v e comma y over 2 equals open parentheses 2 square root of 3 r minus square root of 3 r close parentheses rightwards double arrow y over 2 equals square root of 3 r rightwards double arrow y equals 2 square root of 3 r T h u s comma space t h e space p e r i m e t e r space o f space t h e space i s o s c e l e s space t r i a n g l e comma 2 x plus y equals 2 cross times 2 square root of 3 r plus 2 square root of 3 r equals 6 square root of 3 r$