<s>[INST] Hypotenuse on a right triangle in a right triangle, the side $a = 3$ and the subtraction the sides $b - c =\sqrt{3}$, What is the value of the hypotenuse?
Can someone help me? I need understand how to arrive at the answer.
 [/INST] As we assue $a,b,c>0$: We know $b=\sqrt{3}+c>c$, hence we know that $c$ cannot be the hypothenuse (= longes side).
There are only two possibilities left. $b$ is the hypotenuse or $a$ is the hypotenuse.
A) $a^2+c^2=b^2=(c+\sqrt{3})^2 \implies 9 +c^2=c^2+2\sqrt 3c + 3 \implies c = \frac{6}{2\sqrt{3}}$
B) $c^2+(c+\sqrt{3})^2=a^2 \implies c^2+c^2+2\sqrt 3 c+3=9 \implies 2c^2+2\sqrt 3 c-6=0$
From $2c^2+2\sqrt 3 c-6=0 \implies c^2+\sqrt 3 c-3=0 \implies c_{1,2}=\frac{-\sqrt{3}\pm\sqrt{3-4(-3)}}{2}$
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