Question

In \(\Delta ABC\) with the usual notations, angles \(A,B,C\) of are in A.P. and \(b:c=\sqrt{3}:\sqrt{2}\). The ratio of the largest and the smallest angle of the triangle will be

• A. \(2:1\)
• B. \(3:1\)
• C. \(5:3\)
• D. \(4:3\)

Hint:

A.P. angles give B = 60 degrees; then apply the sine rule with b : c.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
If \(A,B,C\) are in A.P., then \(2B=A+C\) and \(3B=180^{\circ}\), so \(B=60^{\circ}\).

Step 2: Sine rule
\(\dfrac{b}{c}=\dfrac{\sin B}{\sin C}=\dfrac{\sqrt{3}}{\sqrt{2}}\).

Step 3: Find C
\(\sin C=\dfrac{\sqrt{2}}{\sqrt{3}}\cdot\dfrac{\sqrt{3}}{2}=\dfrac{1}{\sqrt{2}}\Rightarrow C=45^{\circ}\) (the value \(135^{\circ}\) is impossible as \(60^{\circ}+135^{\circ}>180^{\circ}\)).

Step 4: Find ratio
\(A=180^{\circ}-60^{\circ}-45^{\circ}=75^{\circ}\).
Largest : smallest \(=75:45=5:3\).

Final Answer:
The ratio of the largest to the smallest angle is 5 : 3. \[ \boxed{\text{(C) }5:3} \]