Question

Assertion :If two angles of a triangle are 45${}^o$ and 60${}^o$, then the ratio of the smallest and the greatest side are $(\sqrt{3}-1):1$ Reason: The greatest side of a triangle is opposite to its greatest angle.

• 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 is incorrect but Reason is correct

Answer 1

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

The greatest side of a triangle is opposite to its greatest angle.
Similarly the smallest side of a triangle is opposite to its smallest angle.

Given two angles ${45}^0,{60}^0$ then the third angle $({180}^0-({45}^0+{60}^0)={75}^0$

the smallest angle is ${45}^0$ and the largest angle is ${75}^0$

Hence the required ratio will be

$=\frac{sin\left({45}^0\right)}{sin{75}^0}$
$=\frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{3}+1}{2\sqrt{2}}}$
$=\frac{2}{\sqrt{3}+1}$

Rationalize the denominator

$=\frac{2}{\sqrt{3}+1}\times \frac{\sqrt{3}-1}{\sqrt{3}-1}$

$=\sqrt{3}-1$

Hence both Assertion and Reason are correct and Reason is the correct explanation for Assertion