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Question and Answer

In a triangle \(ABC,\) if \(\sin{A},\,\sin{B},\,\sin{C}\) are in A.P then
(A) altitudes are in A.P
(B) altitudes are in H.P
(C) medians are in G.P
(D) medians are in A.P

Answer

Answer: B
\(\sin{A},\,\sin{B},\,\sin{C}\) are in A.P

\(\Rightarrow a,\,b,\,c\) are in A.P

We know that Area of a triangle \(\Delta=\dfrac{1}{2}a{h}_{1}=\dfrac{1}{2}b{h}_{2}=\dfrac{1}{2}c{h}_{3}\)

\(2\Delta=a{h}_{1}=b{h}_{2}=c{h}_{3}\)

\(\therefore {h}_{1}:{h}_{2}:{h}_{3}\)

\(=\dfrac{2\Delta}{a}=\dfrac{2\Delta}{b}=\dfrac{2\Delta}{c}\)

\(=\dfrac{1}{a}=\dfrac{1}{b}=\dfrac{1}{c}\)

\(\Rightarrow a,\,b,\,c\) are in A.P

\(\Rightarrow {h}_{1},\,{h}_{2},\,{h}_{3}\) are in H.P
Hence proved
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