I see that the hypotenuse of a right triangle is opposite the right angle, but how is it always the longest side? I also know that it connects to endpoints of other sides. Please help me out with this! I"m really wanting to know this surprising thing. Here"s an example of a right triangle:
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This is an isosceles right triangle because sides a and b (the height and the base) are the same lengths with two of the angles being 45 degrees adding up to a total with the right angle of 180 degrees (all triangles have angles that add up to 180 degrees). I just want to know from this triangle or any other right triangles why the hypotenuse is the longest side. You"ll really be helping me out.




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asked Oct 26 "14 at 2:17
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Let $a$ be the hypotenuse and $b,c$ the others sides, then by Pythagorean Theorem$$a^2=b^2+c^2.$$Then$$a^2>b^2,quad a^2>c^2.$$Therefore$$a>bquad a>c.$$


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answered Oct 26 "14 at 2:20
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Alternatively, you have the law of sines: For any triangle with sides, $A,B,C$ and corresponding angles $a,b,c$ with angle $a$ opposite side $A$ et cetera, you have the following:

$$fracsin aA = fracsin bB = fracsin cC$$

Let $a$ be $90$ degrees, making $A$ our hypotenuse. Since $a+b+c = 180$ and we don"t want to consider negative angles or angles equal to zero in a triangle, we have that $b



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