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Two sides of a triangle are given by the roots of the equation $x^{2} - 5 x + 6 = 0$ and the angle between the sides is $\frac{π}{3}$ . Then the perimeter of the triangle is

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$5 + \sqrt{2}$

$5 + \sqrt{3}$

$5 + \sqrt{5}$

$5 + \sqrt{7}$

answer is D.

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Detailed Solution

$x^{2} - 5 x + 6 = 0 \Rightarrow x^{2} - 2 x - 3 x + 6 = 0$

$\Rightarrow (x - 2) (x - 3) = 0$

$x = 2, 3 \Rightarrow a = 2, b = 3$

$c^{2} = a^{2} + b^{2} - 2 a b \cos c$

$c^{2} = 4 + 9 - 12 \times \cos \frac{π}{3} = 13 - 12 \times \frac{1}{2} = 7$

$c = \sqrt{7}$

Perimeter of triangle $= 5 + \sqrt{7}$

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