if the angles of a triangle are in the... - Solved

Question 41

Mathematics Trigonometric Equations Hard

if the angles of a triangle are in the ratio 2 : 3 : 7, then the ratio of the sides opposite to these angles is

(A) \(\sqrt{2} : 2 : \sqrt{3} + 1\)

(B) \(2 : \sqrt{2} : \sqrt{3} + 1\)

(C) \(2 : \sqrt{2} : \dfrac{\sqrt{2}}{\sqrt{3}-1}\)

(D) \(\dfrac{1}{\sqrt{2}} : 2 : \dfrac{\sqrt{3}+1}{2}\)

✓ Correct Answer: Option A Verified Solution

Angles in the ratio 2 : 3 : 7 add to 180°, so they are 30°, 45° and 105°.

By the sine rule, sides are in the ratio \(\sin 30° : \sin 45° : \sin 105° = \frac{1}{2} : \frac{1}{\sqrt{2}} : \frac{\sqrt{3}+1}{2\sqrt{2}}\)

Multiply each by \(2\sqrt{2}\): \(\sqrt{2} : 2 : \sqrt{3} + 1\)

Answer: (A)

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