MHT CET · Maths · Properties of Triangles
In , with usual notations, if are in then
- A
- B
- C
- D
Answer & Solution
Correct Answer
(C)
Step-by-step Solution
Detailed explanation
Since \(a, b, c\) are in \(A . P . \Rightarrow 2 b= a + c\)
\(=\frac{ a }{2}\left(2 \cos ^2\left(\frac{ C }{2}\right)\right)+\frac{ c }{2}\left(2 \cos ^2\left(\frac{A}{2}\right)\right)\)
\(=\frac{ a }{2}(1+\cos C )+\frac{ c }{2}(1+\cos A )\)
\(=\frac{1}{2}( a +\operatorname{acos} C + c + c \cos A)\)
\(=\frac{1}{2}( a + c + b )\) (using projection formula)
\(=\frac{1}{2}(2 b+b) \quad(\because a, b, c\) are in A.P. \()\)
\(=\frac{3 b}{2}\)
\(=\frac{ a }{2}\left(2 \cos ^2\left(\frac{ C }{2}\right)\right)+\frac{ c }{2}\left(2 \cos ^2\left(\frac{A}{2}\right)\right)\)
\(=\frac{ a }{2}(1+\cos C )+\frac{ c }{2}(1+\cos A )\)
\(=\frac{1}{2}( a +\operatorname{acos} C + c + c \cos A)\)
\(=\frac{1}{2}( a + c + b )\) (using projection formula)
\(=\frac{1}{2}(2 b+b) \quad(\because a, b, c\) are in A.P. \()\)
\(=\frac{3 b}{2}\)
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