Trigonometric Equations Question 105
Question: In a triangle, the length of the two larger sides are 10 cm and 9 cm respectively. If the angles of the triangle are in A.P., then the length of the third side in cm can be
[MP PET 1990, 2001; IIT 1987; DCE 2001]
Options:
A) $ 5-\sqrt{6} $ only
B) $ 5+\sqrt{6} $ only
C) $ 5-\sqrt{6} $ or $ 5+\sqrt{6} $
D) Neither $ 5-\sqrt{6} $ nor $ 5+\sqrt{6} $
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Answer:
Correct Answer: C
Solution:
- We know that in triangle larger the side larger the angle. Since angles $ \angle A,,\angle B $ and $ \angle C $ are in AP. Hence $ \angle B=60^{o} $ $ \cos B=\frac{a^{2}+c^{2}-b^{2}}{2ac}\Rightarrow \frac{1}{2}=\frac{100+a^{2}-81}{20a} $
$ \Rightarrow $ $ a^{2}+19=10a\Rightarrow a^{2}-10a+19=0 $ $ $ $ a=\frac{10\pm \sqrt{100-76}}{2}\Rightarrow a+c\sqrt{2}=5\pm \sqrt{6} $ .
BETA
