Question

Assertion :In any triangle $ABC,$ if $a:b:c=4:5:6$ then $R:r=16:7$ Reason: In any triangle $ABC:$ $\frac{R}{r}=\frac{abc}{4s}$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$a:b:c=4:5:6a=kb=kc=k$

$\cos A=\frac{b^2+c^2-a^2}{2bc}=\frac{25k^2+36k^2-16k^2}{2.5k.6k}$

$\cos A=\frac{3}{4},\sin A=\sqrt{1-{\cos }^{2}A}=\sqrt{1-\frac{9}{16}}=\frac{\sqrt{7}}{4}$

$R=\frac{abc}{4△},r=\left(\frac{△}{S}\right),△=\frac{R}{r}bc\sin \left(\frac{A}{r}\right)=15k^2\sin A$

$s=\frac{(a+b+c)}{2}=\frac{\left(15k\right)}{2},\frac{R}{r}=\frac{s\times a}{bc{\sin }^{2}A}=\frac{\left(\frac{15k}{2}\right).4k\times 16}{5k.6k.7}$

$\frac{R}{r}=\frac{16}{7},\frac{R}{r}\ne \frac{abc}{4s}$