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Chapter 19: Problem 1867

If \(a, b, c\) the sides of \(\Delta A B C\) are in A.P. and a is the smallest side then cosA equals (a) \([(3 c-4 b) / 2 c]\) (b) \([(3 c-4 b) / 2 b]\) (c) \([(4 c-3 b) / 2 c]\) (d) None of these

Short Answer

Expert verified
The correct answer is (d) None of these.

Step by step solution

01

Write the conditions for the sides in arithmetic progression

As the sides a, b and c are in arithmetic progression (AP) with a being the smallest side, we can write the sides as: \(a, a+d, a+2d\), where d is the common difference. In this case, b = a+d and c = a+2d.
02

Apply the Cosine Rule

We'll use the cosine rule to find the value of cosA: cosA = \(\frac{b^2 + c^2 - a^2}{2bc}\)
03

Substitute the expressions for b and c

Now, substitute the expressions for b and c (from step 1) into the cosine rule (from step 2): cosA = \(\frac{(a+d)^2 + (a+2d)^2 - a^2}{2(a+d)(a+2d)}\)
04

Simplify the expression

Expand and simplify the expression obtained in step 3: cosA = \(\frac{a^2 + 2ad + d^2 + a^2 + 4ad + 4d^2 - a^2}{2(a+d)(a+2d)}\) cosA = \(\frac{a^2 + 6ad + 5d^2}{2(a+d)(a+2d)}\)
05

Compare the simplified expression with the given options

Now, we will compare the simplified expression with the given options: (a) \([(3 c-4 b) / 2 c]\) = \(\frac{(3(a+2d) - 4(a+d))}{2(a+2d)}\) = \(\frac{2d}{2(a+2d)}\) ≠ cosA (b) \([(3 c-4 b) / 2 b]\) = \(\frac{(3(a+2d) - 4(a+d))}{2(a+d)}\) = \(\frac{2d}{2(a+d)}\) ≠ cosA (c) \([(4 c-3 b) / 2 c]\) = \(\frac{(4(a+2d) - 3(a+d))}{2(a+2d)}\) = \(\frac{5d}{2(a+2d)}\) ≠ cosA After looking at all the options provided, none of them match the simplified expression derived in step 4.
06

Conclusion

Thus, the correct answer to this problem is: (d) None of these

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