In Fig 7.59, check whether AD is the bisector of $∠ A$ of $Δ A B C$ in each of the following:
(i) AB =5 cm,AC= 10cm, BD= 1.5 cm and CD=3.5ccm
(ii) AB = 4 cm. AC = 6 cm, BD =1.6cm and CD = 2.4 cm
(iii) AB =8 cm, AC=24 cm, BD=6cm and BC = 24cm
(iv) AB=6cm,AC=8 cm, BD=1.5 cm and CD = 2cm
(v) AB= 5 cm, AC= 12cm, BD = 2.5cm and BC = 9 cm

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Solution

$left parenthesis i right parenthesis space I t space i s space g i v e n space t h a t space A B space equals space 5 space c m comma space A C space equals space 10 space c m comma space B D space equals space 1.5 space c m space a n d space C D space equals space 3.5 space c m W e space h a v e space t o space c h e c k space w h e t h e r space A D space i s space b i s e c t o r space o f space angle space A F i r s t space w e space w i l l space c h e c k space p r o p o r t i o n a l space r a t i o space b e t w e e n space s i d e s. N o w comma fraction numerator A B over denominator A C end fraction equals 5 over 10 equals 1 half fraction numerator B D over denominator C D end fraction equals fraction numerator 1.5 over denominator 3.50 end fraction equals 3 over 7 S i n c e comma space fraction numerator A B over denominator A C end fraction not equal to fraction numerator B D over denominator C D end fraction H e n c e comma space A D space i s space n o t space t h e space b i s e c t o r space o f space angle space A.$

$left parenthesis i i right parenthesis space I t space i s space g i v e n space t h a t space A B space equals space 4 space c m comma space A C space equals space 6 space c m comma space B D space equals space 1.6 space c m space a n d space C D space equals space 2.4 space c m. W e space h a v e space t o space c h e c k space w h e t h e r space A D space i s space t h e space b i s e c t o r space o f space angle space A F i r s t space w e space w i l l space c h e c k space p r o p o r t i o n a l space r a t i o space b e t w e e n space s i d e s. S o comma fraction numerator A B over denominator A C end fraction equals 4 over 6 equals 2 over 3 fraction numerator B D over denominator C D end fraction equals fraction numerator 1.6 over denominator 2.4 end fraction equals 2 over 3 left parenthesis i t space i s space p r o p o r t i o n a l right parenthesis space space H e n c e comma space A D space i s space t h e space b i s e c t o r space o f space angle space A.$

$left parenthesis i i i right parenthesis space I t space i s space g i v e n space t h a t space A B space equals space 8 space c m comma space A C space equals space 24 space c m comma space B D space equals space 6 space c m space a n d space B C space equals space 24 space c m. W e space h a v e space t o space c h e c k space w h e t h e r space A D space i s space t h e space b i s e c t o r space o f space angle space A space F i r s t space w e space w i l l space c h e c k space p r o p o r t i o n a l space r a t i o space b e t w e e n space s i d e s. D C space equals space B C space â�� space B D space D C space equals space 24 space â�� space 6 space space D C space equals space 18 S o comma fraction numerator A B over denominator A C end fraction equals 8 over 24 equals 1 third fraction numerator B D over denominator C D end fraction equals 6 over 18 equals 1 third left parenthesis i t space i s space p r o p o r t i o n a l right parenthesis space space H e n c e comma space A D space i s space t h e space b i s e c t o r space o f space angle space A.$

$left parenthesis i v right parenthesis space A B equals 6 c m comma A C equals 8 space c m comma space B D equals 1.5 space c m space a n d space C D space equals space 2 c m W e space h a v e space t o space c h e c k space w h e t h e r space A D space i s space t h e space b i s e c t o r space o f space angle space A F i r s t space w e space w i l l space c h e c k space p r o p o r t i o n a l space r a t i o space b e t w e e n space s i d e s. S o comma fraction numerator A B over denominator A C end fraction equals 6 over 8 equals 3 over 4 fraction numerator B D over denominator C D end fraction equals fraction numerator 1.5 over denominator 2.0 end fraction equals 3 over 4 left parenthesis i t space i s space p r o p o r t i o n a l right parenthesis space space H e n c e comma space A D space i s space t h e space b i s e c t o r space o f space angle space A.$

$left parenthesis v right parenthesis space A B equals space space 5 space c m comma space A C equals space 12 c m comma space B D space equals space 2.5 c m space a n d space B C space space equals space 9 space c m W e space h a v e space t o space c h e c k space w h e t h e r space A D space i s space b i s e c t o r space o f space angle space A F i r s t space w e space w i l l space c h e c k space p r o p o r t i o n a l space r a t i o space b e t w e e n space s i d e s. N o w comma fraction numerator A B over denominator A C end fraction equals 5 over 12 fraction numerator B D over denominator C D end fraction equals fraction numerator 2.5 over denominator 9 end fraction equals 5 over 18 S i n c e comma space fraction numerator A B over denominator A C end fraction not equal to fraction numerator B D over denominator C D end fraction H e n c e comma space A D space i s space n o t space t h e space b i s e c t o r space o f space angle space A.$

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Similar questions

In a $Δ A B C$ , AD is the bisector of $∠ A$ , meeting side BC at D.
(i) If BD= 2.5 cm, AB = 5 cm and AC = 4.2 cm, find DC.
(ii) If BD= 2 cm, AB = 5 cm and DC = 3 cm, find AC.
(iii) If AB= 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD.
(iv) If AB =10 cm, AC =14 cm and BC = 6 cm, find BD and DC.
(v) If AC = 4.2 cm, DC = 6 cm and BC = 10 cm, find AB.
(vi) If AD= 5.6 cm, AC = 6 cm and DC = 3 cm, find BC.
(vii) If AB= 3.6 cm, BC e 6 cm and BD = 32 cm, find AC.
(viii) If AB =10 cm, AC = 6 cm and BC = 12 cm, find BD and DC