If the sides a, b, c of the triangle A B C are in harmonic progression, then ² A/2, ² B/2, ² C/2 are in [AP EAPCET 22 May 2025 S2]

Correct Answer is : Arithmetic progression

Arithmetico-geometric progression

Correct Answer is : Arithmetic progression

Test Index

Properties of Triangles

Section: Mathematics

Question : 12 of 50

Marks: +1, -0

If the sides

a, b, c

A B C

\csc^2 \frac{A}{2}, \csc^2 \frac{B}{2}, \csc^2 \frac{C}{2}

Solution:

a, b, c

are in HP.

\Rightarrow b=\frac{2ac}{a+c}

Let

\csc^2 \frac{A}{2}, \csc^2 \frac{B}{2}, \csc^2 \frac{C}{2}

are in AP .

\Rightarrow \frac{1}{\sin^2 \frac{A}{2}}, \frac{1}{\sin^2 \frac{B}{2}}, \frac{1}{\sin^2 \frac{C}{2}} \text{ are in AP.}

\Rightarrow \frac{bc}{(s-b)(s-c)}, \frac{ac}{(s-a)(s-c)}

\frac{ab}{(s-a)(s-b)}

are in AP.

\Rightarrow \frac{bc}{s(s-b)(s-c)}, \frac{ac}{s(s-a)(s-c)}

\frac{ab}{s(s-a)(s-b)}

are in AP.

\Rightarrow \frac{bc(s-a)}{s(s-a)(s-b)(s-c)}, \frac{ac(s-b)}{s(s-a)(s-b)(s-c)^{\prime}}, \frac{ab(s-c)}{s(s-a)(s-b)(s-c)}

are in AP.

\Rightarrow b c s - a b c, a c s - a b c, a b s - a b c

are in AP.

\Rightarrow b c s, a c s, a b s

are in AP.

\Rightarrow b c, a c, a b

are in AP.

\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}

are in AP.

\Rightarrow a, b, c

are in HP. Which is true.

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