The sides $a, b, c$ of a triangle satisfy the relations $c^{2} = 2 a b$ and $a^{2} + c^{2} = 3 b^{2} .$ Then the measure of $∠ B A C,$ in degrees, is

30

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45

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60

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90

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Solution

The correct option is B $45$
GIven that $c^{2} = 2 a b,$ and $a^{2} + c^{2} = 3 b^{2}$
$\Rightarrow a^{2} + 2 a b = 3 b^{2} \Rightarrow (a + b)^{2} = 4 b^{2} = (2 b)^{2} \Rightarrow a = b$

$∴ c^{2} = 2 a^{2} or c^{2} = a^{2} + b^{2}$
So, $Δ A B C$ is a right-angled isosceles triangle.
$\Rightarrow ∠ B A C = 45^{\circ}$

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