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Byju's Answer
Standard X
Mathematics
Criteria for Similarity of Triangles
In a Δ ABC, A...
Question

In a ΔABC, 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

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Solution


left parenthesis i right parenthesis I t space i s space g i v e n space t h a t space B D space equals space 2.5 space c m comma space A B space equals space 5 space c m comma space a n d space A C space equals space 4.2 space c m. I n space capital delta space A B C comma space 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 comma space m e e t i n g space s i d e space B C space a t space D. W e space n e e d space t o space f i n d space D C comma S i n c e comma space A D space i s space angle space A space b i s e c t o r comma T h e n comma space space fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction fraction numerator A B over denominator A C end fraction equals fraction numerator 2.5 over denominator D C end fraction fraction numerator 5 over denominator 4.2 end fraction equals fraction numerator 2.5 over denominator D C end fraction rightwards double arrow D C equals 2.1 c m

left parenthesis i i right parenthesis space I t space i s space g i v e n space t h a t space B D space equals space 2 space c m comma space A B space equals space 5 space c m comma space a n d space D C space equals space 3 space c m n space capital delta space A B C 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 comma space m e e t i n g space s i d e space B C space a t space D W e space n e e d space t o space f i n d space A C. S i n c e comma space A D space i s space angle A space b i s e c t o r. T h e n comma space space fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction left parenthesis sin c e 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 a n d space s i d e space B C right parenthesis fraction numerator 5 over denominator A C end fraction equals 2 over 3 rightwards double arrow A C equals 7.5 c m

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 3.5 space c m comma space A C space equals space 4.2 space c m comma space a n d space D C space equals space 2.8 space c m I n space capital delta space A B C 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 comma space m e e t i n g space s i d e space B C space a t space D W e space n e e d space t o space f i n d space B D. S i n c e comma space A D space i s space angle A space b i s e c t o r fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction left parenthesis sin 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 space a n d space s i d e space space B C right parenthesis fraction numerator 3.5 over denominator 4.2 end fraction equals fraction numerator B D over denominator 2.8 end fraction rightwards double arrow B D equals fraction numerator 3.5 cross times 2.8 over denominator 4.2 end fraction equals 7 over 3 equals 2.3 c m

left parenthesis i v 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 10 space c m comma space A C space equals space 14 space c m comma space a n d space B C space equals space 6 space c m I n space capital delta capital delta space A B C 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 space m e e t i n g space s i d e space B C space a t space D W e space n e e d space t o space f i n d space B D space a n d space D C. S i n c e comma space A D space i s space b i s e c t o r space o f space angle A T h e r e f o r e comma fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction left parenthesis A D space i s space b i s e c t o r space o f space angle space A space a n d space s i d e space B C right parenthesis 10 over 14 equals fraction numerator x over denominator 6 minus x end fraction 14 x space equals space 60 space � space 6 x 20 x space equals space 60 x space equals space 60 divided by 20 B D space equals space 3 space c m space a n d space D C space equals space 3 space c m.

left parenthesis v right parenthesis space I t space i s space g i v e n space t h a t space A C space equals space 4.2 space c m comma space D C space equals space 6 space c m comma space a n d space B C space equals space 10 space c m. I n space capital delta space A B C comma space A D space i s space t h e space b i s e c t o r space o f space space angle space A comma space m e e t i n g space s i d e space B C space a t space D. W e space n e e d space t o space f i n d space o u t space A B comma S i 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 T h e r e f o r e comma fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator C D end fraction fraction numerator 4.2 over denominator A B end fraction equals 6 over 4 6 A B space equals space 4.2 space x space 4 A B space equals space left parenthesis 4.2 space x space 4 right parenthesis divided by 6 A B space equals space 16.8 divided by 6 space space A B space equals space 2.8 space c m

left parenthesis v 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 5.6 space c m comma space B C space equals space 6 space c m comma space a n d space D C space equals space 3 space c m I n space capital delta space A B C comma space space A D space i s space t h e space b i s e c t o r space o f space space angle space A comma space space m e e t i n g space s i d e space B C space a t space D W e space n e e d space t o space f i n d space B C comma S i n c e comma space A D space i s space t h e space angle A space b i s e c t o r T h e r e f o r e comma fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction fraction numerator 6 over denominator 5.6 end fraction equals fraction numerator 3 over denominator D C end fraction D C space equals space 2.8 space c m A n d comma space B C space equals space 2.8 space plus space 3 space space B C space equals space 5.8 space c m


left parenthesis v 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 5.6 space c m comma space B C space equals space 6 space c m comma space a n d space B D space equals space 3.2 space c m I n space capital delta space A B C comma space 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 comma space m e e t i n g space s i d e space B C space a t space D T h e r e f o r e comma fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction fraction numerator 5.6 over denominator A C end fraction equals fraction numerator 3.2 over denominator 2.8 end fraction left parenthesis D C space equals space B C space � space B D right parenthesis A C space equals space left parenthesis 5.6 space x space 2.8 right parenthesis divided by 3.2 space space A C space equals space 4.9 space c m

left parenthesis v 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 10 space c m comma space A C space equals space 6 space c m comma space a n d space B C space equals space 12 space c m I n space capital delta space A B C comma space A D space i s space t h e space angle space A space b i s e c t o r comma space m e e t i n g space s i d e space B C space a t space D. space space W e space n e e d space t o space f i n d space B D space a n d space D C S i n c e comma space A D space i s space b i s e c t o r space o f space angle space A S o comma fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator C D end fraction L e t space B D space equals space x space c m T h e n comma 6 over 10 equals fraction numerator 12 minus x over denominator x end fraction 6 x space equals space 120 space � space 10 x 16 x space equals space 120 x space equals space 120 divided by 16 x space equals space 7.5 N o w comma space D C space equals space 12 space � space B D D C space equals space 12 space � space 7.5 D C space equals space 4.5 space space B D space equals space 7.5 space c m space a n d space D C space equals space 4.5 space c m.


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