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प्रश्न
Mark the correct alternative in each of the following:
If the sides of a triangle are in the ratio \[1: \sqrt{3}: 2\] then the measure of its greatest angle is
पर्याय
\[\frac{\pi}{6}\]
\[\frac{\pi}{3}\]
\[\frac{\pi}{2}\]
\[\frac{2\pi}{3}\]
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उत्तर
Let ∆ABC be the given triangle such that its sides are in the ratio \[1: \sqrt{3}: 2\]
\[\therefore a = k, b = \sqrt{3}k, c = 2k\]
Now,
\[a^2 + b^2 = k^2 + 3 k^2 = 4 k^2 = c^2\]
So, ∆ABC is a right triangle right angled at C.
\[\therefore C = 90°\]
Using sine rule, we have
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
\[ \Rightarrow \frac{k}{\sin A} = \frac{\sqrt{3}k}{\sin B} = \frac{2k}{\sin90°\]
\[ \Rightarrow \sin A = \frac{1}{2} \text{ and } \sin B = \frac{\sqrt{3}}{2}\]
\[ \Rightarrow A = 30° a\text{ and } B = 60°\]
Thus, the measure of its greatest angle is \[\frac{\pi}{2}\]
Hence, the correct answer is option (c).
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