MHT CET · Maths · Properties of Triangles
If in \(\triangle \mathrm{ABC}\), with usual notations, \(a \cdot \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}\), then
- A \(a, b, c\) are in G.P.
- B \(a, b, c\) are in H.P.
- C \(a, b, c\) are in A.P.
- D \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in Arithmetico Geometric Progression
Answer & Solution
Correct Answer
(C) \(a, b, c\) are in A.P.
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \quad a \cdot \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2} \\ & \Rightarrow a\left(\frac{1+\cos C}{2}\right)+c\left(\frac{1+\cos A}{2}\right)=\frac{3 b}{2} \\ & \Rightarrow a+a \cos C+c+c \cos A=3 b \\ & \Rightarrow a+b+c=3 b \quad \cdots[\because b=c \cos A+a \cos C] \\ & \Rightarrow a+c=2 b \\ & \therefore \quad a, b, c \text { are in A.P. }\end{aligned}\)
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