Question

If two angles of \( \triangle ABC \) are \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \), then the ratio of the smallest and greatest side is

• A. \( \sqrt{3} : \sqrt{2} \)
• B. \( (\sqrt{3} - 1) : 1 \)
• C. \( (\sqrt{3} + 1) : (\sqrt{3} - 1) \)
• D. \( (\sqrt{3} + 1) : 1 \)

Hint:

Use the Law of Sines to find the ratio of sides in a triangle when given the angles. The side opposite the smallest angle will be the smallest side.

Answer 1

The Correct Option is B

Step 1: Use the Law of Sines.
Given that the angles of \( \triangle ABC \) are \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \), we can use the Law of Sines to relate the sides of the triangle. The law states that: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \] where \( A, B, C \) are the angles and \( a, b, c \) are the corresponding opposite sides.
Step 2: Find the ratio of the smallest and greatest sides.
By applying the Law of Sines, we find that the ratio of the smallest side to the greatest side is \( (\sqrt{3} - 1) : 1 \). Thus, the correct answer is option (B).