Properties of Triangle
1 JEE Maths previous year questions on Properties of Triangle — options free on every question; 1 include the answer & explanation free, the rest unlock with PYQ Pass.
Let \(\frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)}\), where \(A, B, C\) are angles of a triangle \(A B C\). If the lengths of the sides opposite these angles are \(a, b, c\) respectively, then:
[JEE Main 2021, 27 Aug (Shift 1)]
\(b^2, c^2, a^2\) are in \(A . P\)
\( \frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)} \)
\(\sin A \sin C \cos B-\sin A \cos C \sin B=\sin B \sin A \cos C - \sin B \cos A \sin C \)
\( 2 \sin A \sin B \cos C=\sin A \sin C \cos B+\sin B \sin C \cos A \)
Using sine rule
\(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=k\\ 2⋅ak⋅bk⋅(\frac{{a}^{2}+{b}^{2}−{c}^{2}}{2ab})=ak⋅ck⋅(\frac{{c}^{2}+{a}^{2}−{b}^{2}}{2ac})+bk\cdot ck\left(\frac{{b}^{2}+{c}^{2}-{a}^{2}}{2bc}\right)\\ \Rightarrow 2({a}^{2}+{b}^{2}−{c}^{2})={c}^{2}+{a}^{2}−{b}^{2}+{b}^{2}+{c}^{2}−{a}^{2}\\ \Rightarrow 2{c}^{2}={a}^{2}+{b}^{2}\)
\(\Rightarrow {b}^{2},{c}^{2},{a}^{2}\) are in A.P
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