Question:
If the angles of triangle ABC are in arithmetic progression and the sides a, b and c satisfy \( \frac{\sqrt{3}}{2} < \frac{b}{a} < 1 \) and \( c < b \), then the possible values of the side c are:
If the angles of triangle ABC are in arithmetic progression and the sides a, b and c satisfy \( \frac{\sqrt{3}}{2} < \frac{b}{a} < 1 \) and \( c < b \), then the possible values of the side c are:
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Whenever angles of a triangle are in A.P., remember \( B=60^\circ \) immediately, which simplifies the cosine rule into a quadratic equation in terms of the sides.
Updated On: Jun 9, 2026
- \( \frac{a\pm\sqrt{4b^2-3a^2}}{2a} \)
- \( \frac{a\pm\sqrt{4b^2-3a^2}}{4} \)
- \( \frac{a\pm\sqrt{4b^2-3a^2}}{2b} \)
- \( \frac{a\pm\sqrt{4b^2-3a^2}}{2} \)
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The Correct Option is D
Solution and Explanation
Concept:
If angles \( A, B, C \) are in A.P., then \( B = 60^\circ \). Using the Law of Cosines for \( \cos B \):
\[ \cos B = \frac{a^2+c^2-b^2}{2ac} = \frac{1}{2} \]
Step 1: Simplify the cosine rule equation.
\( a^2 + c^2 - b^2 = ac \Rightarrow c^2 - ac + (a^2 - b^2) = 0 \).
Step 2: Solve for \( c \) using the quadratic formula.
\( c = \frac{-(-a) \pm \sqrt{(-a)^2 - 4(1)(a^2-b^2)}}{2(1)} \) \[ c = \frac{a \pm \sqrt{a^2 - 4a^2 + 4b^2}}{2} \] \[ c = \frac{a \pm \sqrt{4b^2 - 3a^2}}{2} \]
Step 3: Conclusion.
This matches option (D). a4b^2-3a^22
Step 1: Simplify the cosine rule equation.
\( a^2 + c^2 - b^2 = ac \Rightarrow c^2 - ac + (a^2 - b^2) = 0 \).
Step 2: Solve for \( c \) using the quadratic formula.
\( c = \frac{-(-a) \pm \sqrt{(-a)^2 - 4(1)(a^2-b^2)}}{2(1)} \) \[ c = \frac{a \pm \sqrt{a^2 - 4a^2 + 4b^2}}{2} \] \[ c = \frac{a \pm \sqrt{4b^2 - 3a^2}}{2} \]
Step 3: Conclusion.
This matches option (D). a4b^2-3a^22
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