The Converse of the Pythagorean Theorem

1
The Converse of the Pythagorean Theorem

• A Pythagorean triple is a set of three integers
  which satisfy the Pythagorean Theorem
• Some examples are
• 3-4-5 5-12-13 7-24-25 8-15-17
• Multiples of Pythagorean triples are also
  triples
• 6-8-10 10-24-26 14-48-50 16-30-34
• 9-12-15
• 12-16-20
• C-83 Converse of the Pythagorean Theorem
• If the lengths of the three sides of a triangle
  satisfy the Pythagorean equation, then the
  triangle is a right triangle

hypotenuse
legs
2
The Converse of the Pythagorean Theorem

• Proving the Converse of the Pythagorean Theorem
• Given ?ABC with side lengths such that a2 b2
  c2
• Construct right ?DEF with sides of length a and
  b, and hypotenuse of length x
• By the Pythagorean Theorem, a2 b2 x2
• By substitution, x2 c2, so x c
• The two triangles are congruent by SSS
• Angle C must be a right angle by CPCTC
• Therefore ?ABC is a right triangle, QED

3
Right Triangle Congruence

• Prove that two right triangles with one leg and
  hypotenuse congruent are congruent
• Given right ?ABC and right ?DEF with legs of
  length a and hypotenuse of length c
• Is there a congruence rule that proves ?ABC _at_
  ?DEF?
• The matching parts would be SSA, which is not a
  congruence rule
• By the Pythagorean Theorem, a2 b2 c2, so the
  other legs of the two triangles must also be
  congruent
• Therefore the triangles are congruent by SSS
• Hypotenuse-Leg Congruence Theorem
• If one leg and the hypotenuse of one right
  triangle is congruent to one leg and the
  hypotenuse of a second right triangle, then the
  triangles are congruent

c