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Bharat Dani
Asked: 2022-11-10T00:46:09+05:30 In:

If a, b, c are in GP and log a – log 2b, log 2b – log 3c and log 3c – log a are in AP, then a, b, c are the lengths of the sides of a triangle which is
1. Acute angle
2. Obtuse angled
3. Right angles
4. Equilateral

If a, b, c are in GP and log a – log 2b, log 2b – log 3c and log 3c – log a are in AP, then a, b, c are the lengths of the sides of a triangle which is
1. Acute angle
2. Obtuse angled
3. Right angles
4. Equilateral
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1 Answer

  1. Correct Answer – Option 2 : Obtuse angled

    Concept:         

    The sides of an obtuse triangle should satisfy the condition that the sum of the square of two smaller sides should be less than the square of the largest side.

    If a, b, c are in AP than \(b = \frac{{a + c}}{2}\)

    If a, b, c are in GP than \(b = \sqrt {ac} \)

    Calculation:
     log a – log 2b, log 2b – log 3c and log 3c – log a are in AP
    \(\log 2b – \log 3c = \frac{{\log 3c – \log a + \log a – \log 2b}}{2}\)
    \(\log \frac{{2b}}{{3c}} = \frac{{\log \frac{{3c}}{{2b}}}}{2}\)
    \(\frac{{4{b^2}}}{{9{c^2}}} = \frac{{3c}}{{2b}}\)
    2b = 3c   ———–(1)
    ∵ a, b, c are in GP
    ⇒ b2 = ac  ———–(2)
    From equation 1 and 2 we get
    9c = 4a
    a:b:c = 9:2:4
    As we know that, for obtuse triangle  the sum of the square of two smaller sides should be less than the square of the largest side.
    ⇒ a2 > b2 + c2
    So, a, b and c are the sides of an obtuse angle triangle.
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