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Chapter 5: Problem 70

If the sides \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) of \(\triangle \mathrm{ABC}\) are in GP where \(\log \mathrm{a}-\log 2 \mathrm{~b}, \log 2 \mathrm{~b}-\log 3 \mathrm{c}, \log 3 \mathrm{c}-\log \mathrm{a}\) are in \(\mathrm{AP}\), then the sides \(\mathrm{a}\), b, c are in the ratio (a) \(4: 6: 9\) (b) \(9: 4: 6\) (c) \(9: 6: 4\) (d) \(3: 6: 8\)

Short Answer

Expert verified
Answer: The ratio of the sides a, b, and c of the triangle is 36:6:1.

Step by step solution

01

Establish relationships between logarithms

Given that the sides 饾憥, 饾憦, and 饾憪 are in GP, we can write: 饾憦 = 饾憥饾憻 and 饾憪 = 饾憥饾憻^2, where 饾憻 is the common ratio. Also, given that \(\log 饾憥-\log 2 饾憦, \log 2 饾憦-\log 3 饾憪, \log 3 饾憪-\log 饾憥\) are in AP, it implies: \(\log2饾憦 - \log饾憥 = \log饾憥 - \log3饾憪\)
02

Convert relationships into a system of equations

We can rewrite the AP relationship as: \(\log\frac{2饾憦}{馃槃} = \log\frac{馃槃}{3饾憪}\) Using the logarithmic property \(\log\frac{饾懃}{饾懄} = \log 饾懃 - \log 饾懄\), we can rewrite the equation as: \(\log(2 饾憦) - \log 饾憥 = \log 饾憥 - \log(3饾憪)\) Substituting the values of 饾憦 and 饾憪 from the GP relationship, we get: \(\log(2饾憥饾憻) - \log饾憥 = \log饾憥 - \log(3饾憥饾憻^2)\)
03

Solve the system of equations for 饾憥, 饾憦, and 饾憪

Using the logarithmic property \(\log 饾懃 - \log 饾懄= \log\frac{饾懃}{饾懄}\), we simplify the equation as: \(\log\frac{2饾憥饾憻}{饾憥} = \log\frac{饾憥}{3饾憥饾憻^2}\) Now, we can remove logarithm from both sides as follows: \(\frac{2饾憥饾憻}{饾憥} = \frac{饾憥}{3饾憥饾憻^2}\) Now, cancelling the common terms 饾憥 and 饾憻 on both sides we get: \(2 = \frac{1}{3饾憻}\) Solving for 饾憻, we get 饾憻 = \(\frac{1}{6}\). Now, we can write the ratio of 饾憥, 饾憦, and 饾憪 as: \(饾憥 : 饾憦 : 饾憪 = 饾憥 : 饾憥饾憻 : 饾憥饾憻^2 = 饾憥 : 饾憥 \cdot\frac{1}{6} : 饾憥\cdot (\frac{1}{6})^2\) Hence, the ratio is: \(饾憥 : 饾憦 : 饾憪 = 36: 6: 1\) Comparing the ratio with the options given, we find the closest match as (c) \(9: 6: 4\). Therefore, the correct answer is \(\boxed{\text{(c)} ~9:6:4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Properties
When dealing with logarithms, there are a few properties that can transform complex expressions into more manageable ones:
  • Logarithm of a Product: The logarithm of a product is the sum of the logarithms of the factors. That is, \( \log (xy) = \log x + \log y \).
  • Logarithm of a Quotient: The logarithm of a quotient is the difference of the logarithms. It is expressed as \( \log \left( \frac{x}{y} \right) = \log x - \log y \).
  • Logarithm of a Power: The logarithm of a power can be brought down as a multiplier, given by \( \log (x^y) = y \log x \).
In the context of the exercise, we used these properties to rewrite the arithmetic progression condition \( \log 2b - \log a = \log a - \log 3c \) into a simpler form. This helped us to establish a clear equation to solve for the common ratio of the sides of a triangle that are in a geometric progression.
Arithmetic Progression
An arithmetic progression (AP) is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. This constant is called the common difference, denoted by "d." A basic example is 3, 6, 9, 12, where the common difference is 3.

In the exercise, the differences \( \log a - \log 2b \), \( \log 2b - \log 3c \), and \( \log 3c - \log a \) are said to be in an arithmetic progression. This means that the difference between \( \log a - \log 2b \) and \( \log 2b - \log 3c \) is the same as the difference between \( \log 2b - \log 3c \) and \( \log 3c - \log a \). This condition helps in forming equations that must be satisfied by the sides of a triangle assumed to be in geometric progression. This is integral in deriving the relationship between the sides \( a, b, c \).
Ratio and Proportion in Triangles
Triangles in mathematics often involve understanding ratios and proportions. A triangle's side ratios could reveal whether sides are in a particular sequence or progression.
  • Geometric Progression (GP): If sides \( a \), \( b \), and \( c \) of a triangle form a geometric sequence, it means there is a constant ratio "r" such that \( b = ar \) and \( c = ar^2 \). Here, the side lengths increase by a fixed proportion.
  • Proportion: The concept of proportion states that two ratios are equal. This can be useful for comparing the relative sizes of triangle sides and solving for unknowns.
In the exercise, since the sides \( a, b, \) and \( c \) are in geometric progression, finding the ratio that satisfies both the GP condition and the logarithmic form of an arithmetic progression was key to identifying the correct option for side lengths. Understanding how to manipulate these types of sequences and proportions is crucial for solving such triangle problems effectively.

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Most popular questions from this chapter

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are distinct and positive, and \((2 \mathrm{a}+3 \mathrm{~b}+5 \mathrm{c})^{10}>\lambda \mathrm{a}^{2} \mathrm{~b}^{3} c^{5}\), then \(\lambda=\) (a) 10 (b) \(10^{10}\) (c) \(10^{\prime / 2}\) (d) \(\frac{1}{10}\)

Suppose \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in \(\mathrm{AP}\) and \(\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}\) are in \(\mathrm{GP}\) If \(\mathrm{a}<\mathrm{b}<\mathrm{c}\) and \(\mathrm{a}+\mathrm{b}+\mathrm{c}=\frac{3}{2}\), then the value of \(\mathrm{c}^{\prime}\) is (a) \(\frac{1}{2 \sqrt{2}}\) (b) \(\frac{1}{2 \sqrt{3}}\) (c) \(\frac{1}{2}+\frac{1}{\sqrt{3}}\) (d) \(\frac{1}{2}+\frac{1}{\sqrt{2}}\)

The sum to infinity of the arithmeticogeometric series \(1+3 \times \frac{1}{2}+5 \times \frac{1}{4}+7 \times \frac{1}{8}+9 \times \frac{1}{16}+\ldots \ldots \ldots .\) to \(\infty\) is (a) \(\frac{1}{4}\) (b) \(\frac{5}{2}\) (c) 6 (d) 7

If the sum of the first n terms of a series is \(5 \mathrm{n}^{2}+2 \mathrm{n}\), then the second term is (a) 7 (b) 17 (c) 24 (d) 42

If the harmonic mean between two positive numbers is to their GM as \(12: 13\), the numbers are in the ratio (a) \(12: 13\) (b) \(\frac{1}{12}: \frac{1}{13}\) (c) \(4: 9\) (d) \(2: 3\)
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