Q67 / 160 · AP EAMCET 2022 (05 Jul Shift 2) · Mathematics

In a triangle, if $b=5, c=6, \tan \frac{A}{2}=\frac{1}{\sqrt{2}}$, then $a=$

  • Correct answer
    A
    $\sqrt{41}$
  • B
    $\sqrt{21}$
  • C
    $\sqrt{14}$
  • D
    $8 \sqrt{6}$

Solution

$\because \cos A=\frac{1-\tan ^2 \frac{A}{2}}{1+\tan ^2 \frac{A}{2}}$ $=\frac{1-\left(\frac{1}{\sqrt{2}}\right)^2}{1+\left(\frac{1}{\sqrt{2}}\right)^2}=\frac{1}{3}$ $\cos A=\frac{b^2+c^2-a^2}{2 b c}$ $\frac{1}{3}=\frac{(5)^2+(6)^2-a^2}{2(5)(6)}$ $\begin{aligned} & 20=25+36-a^2 \\ & a^2=41 \\ & a=\sqrt{41}\end{aligned}$

Hence, the correct answer is option: 'A'.

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