A Proof for the
Converse of the Pythagorean Theorem
Converse of the
Pythagorean Theorem:
If the square of the
length of the longest side of a triangle is equal to the sum of the squares of
the lengths of the other two sides, then the triangle is a right triangle.
Proof:
Suppose the triangle is not a right triangle. Label the vertices A, B and C as
pictured. (There are two possibilities for the measure of angle C: less than 90
degrees (left picture) or greater than 90 degrees (right picture).)
Construct
a perpendicular line segment CD as pictured below.
By
the Pythagorean Theorem, BD² = a² + b² = c², and so BD = c. Thus we have
isosceles triangles ACD and ABD. It follows that we have congruent angles CDA =
CAD and BDA = DAB. But this contradicts the apparent inequalities (see picture)
BDA < CDA = CAD < DAB (left picture) or DAB < CAD = CDA < BDA
(right picture).
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