<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] The sides are $3, c, c+\sqrt{3}$. Now consider possible pythagorean pairs.
Either $3$ can be hypotenuse, or $c+\sqrt{3}$


*

*Using 3 as hypotenuse, we get:
$$9 = 2c^2 + 2c\sqrt{3} +3\\
c^2 + c\sqrt{3} -3=0\\
c = \frac{\sqrt 3}{2}$$

*Using $c+\sqrt{3}$ as hypotenuse, you have:
$$c^2 + 2c\sqrt 3+3=c^2+9 \\
c = \frac{1}{\sqrt{3}}$$
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