MHB Why Can't a Right Triangle Have Sides that Add Up to the Area of Squares?

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A right triangle's sides must satisfy the Pythagorean theorem, which states that the sum of the squares of the two shorter sides equals the square of the longest side. In the given problem, the areas of the squares on the triangle's sides are a^2 = 18, b^2 = 7, and c^2 = 27. The calculation shows that 18 + 7 does not equal 27, indicating that the triangle cannot be a right triangle. Additionally, the theorem does not imply that the lengths themselves can be added directly; rather, the squares of the lengths must be compared. Therefore, the confusion arises from misunderstanding the relationship defined by the theorem.
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So I've been working on Pythagorean stuff and it's pretty straight forward but then I got confused over something quite simple.

It's a geometry question so I'll try my best to illustrate the question.

So there's a triangle and we must evaluate whether it's a right angle. The lengths are not provided but each side of the triangle is also the side a square. So if we can find the length of a side from each square we can find out whether the triangle is a right triangle using the Pythagorean theorem.

The areas of the squares are provided.

The problem is the values say
a^2 = 18
b^2 = 7
c^2 = 27

Now the answer is 18 + 7 != 27 so it's not a right angle but I'm looking at this and thinking that if area is side^2 and it's a^2 = 18 then isn't the side a = sqr(18)?

So then I walk into a quagmire of sqr(18) + sqr(7) ?= sqr(27)... So what's wrong with me? I mean, the short answer.
 
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Ziggletooth said:
So I've been working on Pythagorean stuff and it's pretty straight forward but then I got confused over something quite simple.

It's a geometry question so I'll try my best to illustrate the question.

So there's a triangle and we must evaluate whether it's a right angle. The lengths are not provided but each side of the triangle is also the side a square. So if we can find the length of a side from each square we can find out whether the triangle is a right triangle using the Pythagorean theorem.

The areas of the squares are provided.

The problem is the values say
a^2 = 18
b^2 = 7
c^2 = 27

Now the answer is 18 + 7 != 27 so it's not a right angle but I'm looking at this and thinking that if area is side^2 and it's a^2 = 18 then isn't the side a = sqr(18)?

So then I walk into a quagmire of sqr(18) + sqr(7) ?= sqr(27)... So what's wrong with me? I mean, the short answer.

No quagmire at all.

The Pythagorean Theorem states $a^{2} + b^{2} = c^{2}$.
It says nothing of the $a + b = c$ sort. Thus, why does it concern you?
 
In fact, for a, b, c the lengths of the three sides of any triangle, you can't have "a+ b= c". You must have a+ b> c.
 
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