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

[Ziggletooth]

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.

 

[tkhunny]

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?

 

[HOI]

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.