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Maths-

General

Easy

Question

In $straight triangle A B C comma a b c sin space A over 2 sin space B over 2 sin space C over 2 equals$

Δ²
$\frac{Δ^{2}}{s}$
Δ s
$\frac{s}{Δ^{2}}$

hint Hint:

s denotes the semi-perimeter of the triangle ABC, ∆ its area and R the radius of the circle circumscribing the triangle ABC i.e., R is the circum-radius.The angles of the triangle ABC are denoted by A, B, C and the corresponding opposite sides by a, b, c.

The correct answer is: $straight capital delta squared over s$

$straight triangle A B C comma space a b c sin space A over 2 sin space B over 2 sin space C over 2 W e space k n o w space t h a t space comma bold italic I bold italic n bold space bold italic a bold space bold italic t bold italic r bold italic i bold italic a bold italic n bold italic g bold italic l bold italic e bold comma bold space bold italic s bold italic i bold italic d bold italic e bold space bold “ bold italic a bold ” bold space bold italic d bold italic i bold italic v bold italic i bold italic d bold italic e bold italic d bold space bold italic b bold italic y bold space bold italic t bold italic h bold italic e bold space bold sin bold italic e bold space bold italic o bold italic f bold space bold italic a bold italic n bold italic g bold italic l bold italic e bold space bold italic A bold space bold italic i bold italic s bold space bold italic e bold italic q bold italic u bold italic a bold italic l bold space bold italic t bold italic o bold space bold italic t bold italic h bold italic e bold space bold italic s bold italic i bold italic d bold italic e bold space bold “ bold italic b bold ” bold space bold italic d bold italic i bold italic v bold italic i bold italic d bold italic e bold italic d bold space bold italic b bold italic y bold space bold italic t bold italic h bold italic e bold space bold sin bold italic e bold space bold italic o bold italic f bold space bold italic a bold italic n bold italic g bold italic l bold italic e bold space bold italic B bold space bold italic i bold italic s bold space bold italic e bold italic q bold italic u bold italic a bold italic l bold space bold italic t bold italic o bold space bold italic t bold italic h bold italic e bold space bold italic s bold italic i bold italic d bold italic e bold space bold “ bold italic c bold ” bold space bold italic d bold italic i bold italic v bold italic i bold italic d bold italic e bold italic d bold space bold italic b bold italic y bold space bold italic t bold italic h bold italic e bold space bold sin bold italic e bold space bold italic o bold italic f bold space bold italic a bold italic n bold italic g bold italic l bold italic e bold space bold italic C bold. fraction numerator bold a bold space over denominator bold sin bold space bold A end fraction bold space bold space bold equals bold space bold space fraction numerator bold space bold b over denominator bold sin bold space bold B end fraction bold space bold space bold space bold equals bold space bold space fraction numerator bold space bold c over denominator bold sin bold space bold C end fraction bold space bold equals bold 2 bold italic R bold space bold italic A bold italic l bold italic s bold italic o bold space bold italic w bold italic e bold space bold italic k bold italic n bold italic o bold italic w bold comma bold space bold space bold sin fraction numerator bold space bold A over denominator bold 2 end fraction bold space bold equals square root of bold space fraction numerator bold left parenthesis bold s bold minus bold b bold right parenthesis bold left parenthesis bold s bold minus bold c bold right parenthesis over denominator bold bc end fraction end root bold comma bold space bold italic W bold italic h bold italic e bold italic r bold italic e bold space bold italic s bold space bold italic d bold italic e bold italic n bold italic o bold italic t bold italic e bold italic s bold space bold italic t bold italic h bold italic e bold space bold italic s bold italic e bold italic m bold italic i bold minus bold italic p bold italic e bold italic r bold italic i bold italic m bold italic e bold italic t bold italic e bold italic r bold space bold italic o bold italic f bold space bold italic t bold italic h bold italic e bold space bold italic t bold italic r bold italic i bold italic a bold italic n bold italic g bold italic l bold italic e bold space bold italic A bold italic B bold italic C bold. bold space bold italic A bold italic r bold italic e bold italic a bold space bold increment bold space bold equals square root of bold space bold s bold left parenthesis bold s bold minus bold a bold right parenthesis bold left parenthesis bold s bold minus bold b bold right parenthesis bold left parenthesis bold s bold minus bold c bold right parenthesis end root bold space bold comma bold space bold italic W bold italic h bold italic e bold italic r bold italic e bold space bold 2 bold italic s bold equals bold space bold italic a bold plus bold italic b bold plus bold italic c s o comma space space a b c sin space A over 2 sin space B over 2 sin space C over 2 space a b c cross times square root of space fraction numerator left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis over denominator bc end fraction end root cross times square root of space fraction numerator left parenthesis straight s minus c right parenthesis left parenthesis straight s minus a right parenthesis over denominator ca end fraction end root cross times square root of space fraction numerator left parenthesis straight s minus a right parenthesis left parenthesis straight s minus b right parenthesis over denominator ab end fraction end root a b c cross times fraction numerator left parenthesis straight s minus a right parenthesis left parenthesis straight s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator ab c end fraction cross times s over s space bold rightwards double arrow fraction numerator s left parenthesis straight s minus a right parenthesis left parenthesis straight s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s end fraction bold space bold rightwards double arrow bold increment to the power of bold 2 over bold s$

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