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Question

In a capital delta A B C open parentheses fraction numerator a to the power of 2 end exponent over denominator sin invisible function application A end fraction plus fraction numerator b to the power of 2 end exponent over denominator sin invisible function application B end fraction plus fraction numerator c to the power of 2 end exponent over denominator sin invisible function application C end fraction close parentheses times s i n invisible function application fraction numerator A over denominator 2 end fraction s i n invisible function application fraction numerator B over denominator 2 end fraction s i n invisible function application fraction numerator C over denominator 2 end fraction simplifies to

  1. 2 straight capital delta
  2. Δ
  3. straight capital delta over 2
  4. straight capital delta over 4

Hint:

Use sine rule and substitute the values :
fraction numerator a over denominator S i n A end fraction space equals space fraction numerator b over denominator S i n B end fraction space equals space fraction numerator c over denominator S i n C end fraction space equals space 2 R

The correct answer is: Δ


     Given :
    I n space capital delta A B C space open parentheses fraction numerator a squared over denominator sin invisible function application A end fraction plus fraction numerator b squared over denominator sin invisible function application B end fraction plus fraction numerator c squared over denominator sin invisible function application C end fraction close parentheses times s i n invisible function application A over 2 s i n invisible function application B over 2 s i n invisible function application C over 2
    Using sine rule
    a =2RSinA
    b = 2RSinB and
    c = 2RSinC
    Substitute these values
    rightwards double arrow open parentheses fraction numerator 4 R squared S i n squared A over denominator sin invisible function application A end fraction plus fraction numerator 4 R squared S i n squared B over denominator sin invisible function application B end fraction plus fraction numerator 4 R squared S i n squared C over denominator sin invisible function application C end fraction close parentheses times s i n invisible function application A over 2 s i n invisible function application B over 2 s i n invisible function application C over 2

rightwards double arrow space 4 R squared space left parenthesis space S i n A space plus S i n B space plus space S i n C right parenthesis times s i n invisible function application A over 2 s i n invisible function application B over 2 s i n invisible function application C over 2

S i n A space plus S i n B space plus space S i n C space c a n space b e space w r i t t e n space a s space 4 R cos A over 2 cos B over 2 cos C over 2

U sin g space t h i s space f o r m u l a

rightwards double arrow space 16 R squared space left parenthesis space P r o d u c t space o f space a l l space cos A over 2 right parenthesis times left parenthesis P r o d u c t space o f space a l l space sin A over 2 right parenthesis
w e space k n o w space t h a t space S i n 2 A space equals space 2 S i n A C o s A

rightwards double arrow space 2 R squared space left parenthesis 2 space cos A over 2 sin A over 2 times 2 space cos B over 2 sin B over 2.2 space cos C over 2 sin C over 2 right parenthesis
F u r t h e r space s o l v i n g space

rightwards double arrow space 2 R squared space left parenthesis S i n A. S i n B. S i n C right parenthesis

W e space k n o w comma space S i n A. S i n B. S i n C space space equals space fraction numerator a b c over denominator 8 R cubed end fraction
rightwards double arrow space 2 R squared space fraction numerator a b c over denominator 8 R cubed end fraction space equals space fraction numerator a b c over denominator 4 R end fraction space equals triangle

    Thus, in a capital delta A B C open parentheses fraction numerator a to the power of 2 end exponent over denominator sin invisible function application A end fraction plus fraction numerator b to the power of 2 end exponent over denominator sin invisible function application B end fraction plus fraction numerator c to the power of 2 end exponent over denominator sin invisible function application C end fraction close parentheses times s i n invisible function application fraction numerator A over denominator 2 end fraction s i n invisible function application fraction numerator B over denominator 2 end fraction s i n invisible function application fraction numerator C over denominator 2 end fraction simplifies to triangle

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