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ABC is a right triangle with angle B =90 degree.A circle with BC as diameter meets hypotenuse AC at point D.Prove that: (i) AC*AD=AB*AB (ii)BD*BD=AD*DC
Asked by ashutoshtripathy2000 | 02 Oct, 2015, 11:14: AM
answered-by-expert Expert Answer
triangle A B C space i s space a space r i g h t space t r i a n g l e space r i g h t space a n g l e d space a t space B. B C space i s space t h e space d i a m e t e r space o f space t h e space c i r c l e space a n d space D space i s space t h e space p o i n t space w h e r e space t h e space c i r c l e space m e e t s space t h e space l i n e space s e g m e n t space A C A n g l e space i n space a space s e m i space c i r c l e space i s space 90 degree. T h u s comma space angle B D C equals angle B D A equals 90 degree left parenthesis i right parenthesis N o w space c o n s i d e r space t h e space t r i a n g l e s space triangle A B C space a n d space triangle A D B angle A B C equals angle A D B equals 90 degree angle B A C equals angle D A B T h u s space b y space A n g l e minus A n g l e space s i m i l a r i t y comma space w e space h a v e comma triangle A B C space tilde triangle A D B. H e n c e space t h e space c o r r e s p o n d i n g space s i d e s space a r e space p r o p o r t i o n a l. T h u s comma fraction numerator A C over denominator A B end fraction equals fraction numerator A B over denominator A D end fraction rightwards double arrow A C cross times A D equals A B cross times A B left parenthesis i i right parenthesis N o w space c o n s i d e r space t h e space t r i a n g l e s space triangle A D B space a n d space triangle B D C angle A D B equals angle B D C equals 90 degree L e t space angle B C D equals x comma space rightwards double arrow angle C B D equals 90 degree minus x rightwards double arrow angle A B D equals 90 degree minus open parentheses 90 degree minus x close parentheses rightwards double arrow angle A B D equals x equals angle B C D T h u s space b y space A n g l e minus A n g l e space s i m i l a r i t y comma space w e space h a v e comma triangle A D B space tilde triangle B D C. H e n c e space t h e space c o r r e s p o n d i n g space s i d e s space a r e space p r o p o r t i o n a l. T h u s comma fraction numerator A D over denominator B D end fraction equals fraction numerator B D over denominator C D end fraction rightwards double arrow B D cross times B D equals C D cross times A D
Answered by Vimala Ramamurthy | 02 Oct, 2015, 09:58: PM

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