Why is the hypotenuse always positive

if you draw a unit circle the hypotenuse no matter where you put the triangle whether it be the first, second third or fourth quadrant it remains positive, whereas the adjacent and opposite change.

Why do the adjacent and opposite change in different quadrants but not the hypotenuse,

I've seen some people say it is because the hypotenuse is a length meaning it cannot be negative but with that logic wouldn't the opposite and adjacent remain positive?

I've also seen people say it is to do with the Pythagoras theorem $A^2+b^2=C^2$ but that still doesn't seem to make sense since the square root of $C^2$ could be a negative or a positive.

Trigonometry
Math Gnome Math Gnome
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1 Answer

In mathematics, you need to pay attention to the definitions. It all depends on how you define those numbers. If $A$ and $B$ are the lengths of the adjacent and opposite sides, then they must be positive numbers as lengths of line segments are always positive numbers. But if you define them to be the $x$ and $y$ coordinates of the corresponding point on the unit circle, then they could be positive or negative. Hypotenuse is defined as the length of the longest side, and by definition it should be positive, no matter where you are on the unit circle.

Paul F Paul F
199
  • Math Gnome Math Gnome
    0

    So the hypotenuse is always positive because it is defined as the length of the longest side meaning you can't have a negative length, so it must always be positive and with the opposite and adjacent you can define them as coordinates meaning you can get a negative or a positive. Am I understanding that correctly one last thing why is the hypotenuse defined as the longest side and not just the line segment diagonal to the right angle or something like that? Thanks so much for the response.

    • Paul F Paul F
      +1

      Yes, your understanding is correct. You can also define the hypotenuse the way you define it. Both definitions are fine. One can define mathematical objects in several different ways.

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