Why do the special right triangles use values that are irrational instead of rational? Can someone explain this using the theorem of Pythagoras or in a simpler way? Thanks!

Question

anonymous · Answer

I'm not sure what you mean -- how do you define special right triangles?

anonymous · Answer

30, 45 degrees.

anonymous · Answer

The 3-4-5, 5-12-13, etc.. right triangles are right triangles with rational numbers.  I'm not sure what you are asking.

anonymous · Answer

Sorry, what I am trying to figure out the reasoning behind irrational values. Why dont rational values work for some of the triangles sides.

anonymous · Answer

I am still confused what you are confused about.

Do you know what raional numbers and irrational numbers are?

anonymous · Answer

Well based on the Pythagorean theorem,
$$ c = \sqrt{a^2+b^2} $$
where c is the hypotenuse and a and b are the other sides.

There's no reason to think that a^2 and b^2 would happen to come out to be a perfect square.  It's possible, but because a^2 + b^2 cannot be written as (something)^2 in general, you're not going to get a perfect square.   Therefore, you're not going to get a rational number.

anonymous · Answer

Ok, alright I think I get it now.

anonymous · Answer

Thanks.

anonymous · Answer

Ok, the book mentions commensurable. What does this mean. I tried Wikipedia and Google and I still don't understand the term. It is used in this context "The hypotenuse of a right-angles isosceles triangle with the sides of unit length is not commensurable with the unit of length."