Question 415334: This question is from the practice test of the CSET MATH/SCIENCE SUBTEST.
Four congruent triangles, each having legs of length a and b and hypotenuse of length c, are arranged to produce square EFGH.
Using your knowledge of algebra and geometry:
*write an expression for the area of square EFGH in terms of the length of its sides;
*write an expression for the area of square EFGH in terms of the area of its component parts (i.e., four triangles and a square); and
*set these two expressions equal and show that this leads to a proof of the Pythagorean theorem.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
In order for the 4 triangles to form a square, the triangles must be isosceles right triangles. That is to say  and a hypotenuse of one of the right triangles forms one of the sides of the square.
The area of the square is then 
But the area of one of the triangles is  , hence the square, being composed of 4 triangles has an area of 
Since we have established that , can be written  which is to say  . Furthermore, again since ,  .
Since both  and are expressions for the same area, namely square EFGH, we can equate the two expressions:
Which is the Pythagorean Theorem.
Unfortunately, this logic does not strictly prove Pythagora's conjecture. That is because we created a situation at the beginning that required the triangles to be isosceles right triangles. Hence, this proof is only strictly valid for isosceles right triangles and not right triangles in general.
John
My calculator said it, I believe it, that settles it
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