Question

Assertion :In a triangle $R\ge 2r$. Reason: In a triangle $\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\le \frac{1}{8}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion false but Reason is true

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$R\ge 2r\frac{a}{2\sin A}\ge (s-a)\tan \left(\frac{A}{2}\right)\frac{a}{s-a}\ge 2\sin A\tan \frac{A}{2}\frac{a}{(s-a)}\ge 2\times 2\sin \frac{A}{2}\cos \frac{A}{2}\tan \frac{A}{2}4\sin {}^2\frac{A}{2}\le \frac{a}{s-a}$

$\sin {}^2\frac{A}{2}\le \frac{a}{4(s-a)}$

Similarly, $\sin {}^2\frac{B}{2}\le \frac{b}{4(s-b)}\sin {}^2\frac{C}{2}\le \frac{c}{4(s-c)}\sin {}^2\frac{A}{2}\sin {}^2\frac{B}{2}\sin {}^2\frac{C}{2}\le \frac{abc}{64(s-a)(s-b)(s-c)}$

$\therefore \sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}=\frac{1}{8}$

Hence , Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.