(C) Statement $I$: We know that $r = 4R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$. This is a standard identity in a triangle. Thus,$I$ is tr

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(C) Statement $I$: We know that $r = 4R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$. This is a standard identity in a triangle. Thus,$I$ is true.Statement $II$: We know that $r_1 = s \tan \frac{A}{2}$. The given formula $r_1 = (s-a) \tan \frac{A}{2}$ is incorrect. Thus,$II$ is false.Statement $III$: We know that $r_3 = \frac{\Delta}{s-c}$. This is a standard formula for the exradius $r_3$. Thus,$III$ is true.Therefore,statements $I$ and $III$ are correct.

Class 11 Mathematics Trigonometrical Equations Circle connected with triangle

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In a $\triangle ABC$,which of the following formulae are correct?
$I. r = 4R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$
$II. r_1 = (s-a) \tan \frac{A}{2}$
$III. r_3 = \frac{\Delta}{s-c}$

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A
Only $I, II$
B
Only $II, III$
C
Only $I, III$
D
$I, II, III$

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Trigonometrical Equations

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Circle connected with triangle

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In a $\triangle ABC$,which of the following formulae are correct?$I. r = 4R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$$II. r_1 = (s-a) \tan \frac{A}{2}$$III. r_3 = \frac{\Delta}{s-c}$

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In $\Delta ABC$,$AB = AC$. Let $P_1$ denote the incircle of $\Delta ABC$. Circle $P_2$ is tangent to sides $AB$,$AC$ and to circle $P_1$. If the radii of circles $P_1$ and $P_2$ are $2$ and $1$ respectively,then the area of $\Delta ABC$ is:

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In a $\triangle ABC$,which of the following formulae are correct?
$I. r = 4R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$
$II. r_1 = (s-a) \tan \frac{A}{2}$
$III. r_3 = \frac{\Delta}{s-c}$

In $\triangle ABC$,with usual notation,observe the two statements given below :
$(I)$ $r r_1 r_2 r_3 = \Delta^2$
$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$
Which of the following is correct?