How to find the missing length of an obtuse triangle if two sides have a length of 10 one of the angles is 120 degrees?

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How to find the missing length of an obtuse triangle if two sides have a length of 10 one of the angles is 120 degrees?

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Answer 1 · 2015-05-21T08:42:47

The length is $10 \sqrt{3}$ $10sqrt(3)$

To calculate we can use the law of cosines:

if we denote the equal sides of the triangle as $a$ $a$ , and the missing one as $b$ $b$ , we can write:

$b^{2} = a^{2} + a^{2} - 2 a^{2} cos 120$ $b^2=a^2+a^2-2a^2cos120$
$b^{2} = 2 a^{2} - 2 a^{2} cos 120$ $b^2=2a^2-2a^2cos120$
$b^{2} = 2 a^{2} (1 - cos 120)$ $b^2=2a^2(1-cos120)$
$b^{2} = 2 \cdot 10^{2} (1 - cos 120)$ $b^2=2*10^2(1-cos120)$
$b^{2} = 200 (1 - cos (90 + 30))$ $b^2=200(1-cos(90+30))$
$b^{2} = 200 (1 + sin 30)$ $b^2=200(1+sin30)$
$b^{2} = 200 (1 + \frac{1}{2})$ $b^2=200(1+1/2)$
$b^{2} = 300$ $b^2=300$
$b = \sqrt{300} = 10 \sqrt{3}$ $b=sqrt(300)=10sqrt(3)$

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How to find the missing length of an obtuse triangle if two sides have a length of 10 one of the angles is 120 degrees?

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