Question

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

A. $$\frac{R}{r}$$
B. $$\frac{R}{2r}$$
C. $$\frac{r}{R}$$  
D. $$\frac{2r}{R}$$
Answer :   $$\frac{r}{R}$$
Solution :
$$\eqalign{ & \frac{{a\cos A + b\cos B + c\cos C}}{{a + b + c}} = \frac{{2R\sin A\cos A + 2R\sin B\cos B + 2r\sin C\cos C}}{{2s}} \cr & \frac{{a\cos A + b\cos B + c\cos C}}{{a + b + c}} = \frac{R}{{2s}} \cdot \left( {\sin 2A + \sin 2B + \sin 2C} \right) \cr & \frac{{a\cos A + b\cos B + c\cos C}}{{a + b + c}} = \frac{R}{{2s}} \cdot 4\sin A\sin B\sin C \cr & \frac{{a\cos A + b\cos B + c\cos C}}{{a + b + c}} = \frac{{2R}}{s} \cdot \frac{{abc}}{{8{R^3}}} = \frac{{abc}}{{4s{R^2}}}. \cr} $$
But, $$R = \frac{{abc}}{{4\vartriangle }},r = \frac{\vartriangle }{s}.$$    So, the value $$ = \frac{{4\vartriangle R}}{{4 \cdot \frac{\vartriangle }{r} \cdot {R^2}}} = \frac{r}{R}.$$

Releted MCQ Question on
Trigonometry >> Properties and Solutons of Triangle

Releted Question 1

If the bisector of the angle $$P$$ of a triangle $$PQR$$  meets $$QR$$  in $$S,$$ then

A. $$QS = SR$$
B. $$QS : SR = PR : PQ$$
C. $$QS : SR = PQ : PR$$
D. None of these
View Answer
Releted Question 2

From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is 15°. The distance of the boat from the foot of the light house is

A. $$\left( {\frac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}} \right)60\,{\text{metres}}$$
B. $$\left( {\frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}} \right)60\,{\text{metres}}$$
C. $${\left( {\frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}} \right)^2}{\text{metres}}$$
D. none of these
View Answer
Releted Question 3

In a triangle $$ABC,$$  angle $$A$$ is greater than angle $$B.$$ If the measures of angles $$A$$ and $$B$$ satisfy the equation $$3\sin x - 4{\sin ^3}x - k = 0, 0 < k < 1,$$       then the measure of angle $$C$$ is

A. $$\frac{\pi }{3}$$
B. $$\frac{\pi }{2}$$
C. $$\frac{2\pi }{3}$$
D. $$\frac{5\pi }{6}$$
View Answer
Releted Question 4

In a triangle $$ABC,$$  $$\angle B = \frac{\pi }{3}{\text{ and }}\angle C = \frac{\pi }{4}.$$     Let $$D$$ divide $$BC$$  internally in the ratio 1 : 3 then $$\frac{{\sin \angle BAD}}{{\sin \angle CAD}}$$   is equal to

A. $$\frac{1}{{\sqrt 6 }}$$
B. $${\frac{1}{3}}$$
C. $$\frac{1}{{\sqrt 3 }}$$
D. $$\sqrt {\frac{2}{3}} $$
View Answer

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Properties and Solutons of Triangle

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