Questions on Geometry: Pythagorean theorem answered by real tutors!

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Here is why the Pythagorean Theorem can only be used to solve right triangles. The Pythagorean Theorem is with a and b the length of the legs of the triangle and c as the hypotenuse, or the longest side. It has variations too, like and . These three are used to figure out, with given any angles, whether a triangle is right, obtuse or acute (respectively) by plugging in the lengths given. If or holds true, than the triangle is obtuse or acute. But if holds true, than the triangle is right, and it has to be right. You are probably more familiar with this version: . It is only true for right triangles, because of the unique relationship among the three side lengths, which never holds true for any other type of triangle. That is why this variation of it (with the = sign) can only be used for right triangles and right triangles only. Right triangles are the only triangles that make this true.

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Edwin's solution:
I need an explanation of why the Pythatgorean theorem can be used only for right triangles that an 8th grader can understand. What I know is that the Pythagorean Theorem is a special case of the Law of Cosines where the Cos(c) is equal to zero which only happens when c=90 degrees. But I don't think an 8th grader has had the law of cosines yet. I'd really appreciate some help with this. Thanks.

Suppose you had this triangle, which is not necessarily a right triangle:



You only know that  but you don't know necessarily 
that angle  is 90°.

Now you construct another triangle, this time a RIGHT triangle 
with the SAME SIZE legs,  and , but with its hypotenuse
 not given to be equal to :



But we know that the Pythagorean theorem DOES hold true for triangle
DEF because we constructed it so that it is a right triangle with
angle E being a right angle. Therefore



And we are given that



So  because both are equal to .

and we can take positive square roots of both sides:



or 

Now we have all three sides of triangle  equaling 
all three corresponding sides of triangle , so the
two triangles are congruent, and therefore angle 
equals angle E which equals 90°, for they are corresponding
parts of congruent triangles. Therefore  is a
right triangle.

Therefore all we need know is that  to be
able to conclude that triangle  is a right triangle.

So the only time the Pythagorean theorem works is when we
have a right triangle.

Edwin