In a - ABC - R - circumradius and r - inradius- The value of acos A - b cos B - ccos C - a - b - c is equal toA- R - rB- R - 2 rC- r - RD- 2 r - R

The correct option is C r/R a cos A+b cos B+c cos C/a+b+c =2R sin A cos A+2 R sin B cos B+2 r sin C cos C/2s =R/2s.(sin2A+sin2B+sin2C) =R/2s.4sinAsinBsinC ...

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Standard XII

Mathematics

Property 3

In a Δ ABC , ...

Question

In a $Δ$ ABC, R = circumradius and r = inradius. The value of $\frac{a c o s A + b c o s B + c c o s C}{a + b + c}$ is equal to

R/r

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R/2r

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r/R

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2 r/R

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Solution

The correct option is C
r/R

$\frac{a c o s A + b c o s B + c c o s C}{a + b + c} = \frac{2 R s i n A c o s A + 2 R s i n B c o s B + 2 r s i n C c o s C}{2 s} = \frac{R}{2 s} . (s i n 2 A + s i n 2 B + s i n 2 C) = \frac{R}{2 s} .4 s i n A s i n B s i n C = \frac{R}{2 s} .4 \frac{a b c}{8 R^{3}} = \frac{a b c}{4 s R^{2}} B u t R = \frac{a b c}{4 Δ}, r = \frac{Δ}{s} S o, t h e v a l u e = \frac{4 Δ R}{4. \frac{Δ}{r} . R^{2}} = \frac{r}{R}$

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In a Δ ABC, R = circumradius and r = inradius. The value of acosA+bcosB+ccosCa+b+c is equal to

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