If a, b, c are in GP and log a - log 2b, log 2b - log 3c and log 3c -

1.	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 is1. Acute angle2. Obtuse angled3. Right angles4. Equilateral
Answer» 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.

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 is1. Acute angle2. Obtuse angled3. Right angles4. Equilateral

Answer» 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.

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 is1. Acute angle2. Obtuse angled3. Right angles4. Equilateral

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