8-1 The Pythagorean Theorem and Its Converse

1
8-1 The Pythagorean Theorem and Its Converse
2
Parts of a Right Triangle

• In a right triangle, the side opposite the right
  angle is called the hypotenuse.
• It is the longest side.
• The other two sides are called the legs.

3
The Pythagorean Theorem

• Pythagorean Theorem If a triangle is a right
  triangle, then the sum of the squares of the
  lengths of the legs is equal to the square of the
  length of the hypotenuse.
• a2 b2 c2

4
Pythagorean Triples

• A Pythagorean triple is a set of nonzero whole
  numbers that satisfy the Pythagorean Theorem.
• Some common Pythagorean triples include
• 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
• If you multiply each number in the triple by the
  same whole number, the result is another
  Pythagorean triple!

5
Finding the Length of the Hypotenuse

• What is the length of the hypotenuse of ?ABC? Do
  the sides form a Pythagorean triple?

6
? The legs of a right triangle have lengths 10
and 24. What is the length of the hypotenuse?
Do the sides form a Pythagorean triple?
7
Finding the Length of a Leg

• What is the value of x? Express your answer in
  simplest radical form.

8
? The hypotenuse of a right triangle has length
12. One leg has length 6. What is the length of
the other leg? Express your answer in simplest
radical form.
9
Triangle Classifications

• 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.
• If c2 a2 b2, than ?ABC is a right triangle.
• Theorem 8-3 If the square of the length of the
  longest side of a triangle is great than the sum
  of the squares of the lengths of the other two
  sides, then the triangle is obtuse.
• If c2 gt a2 b2, than ?ABC is obtuse.
• Theorem 8-4 If the square of the length of the
  longest side of a triangle is less than the sum
  of the squares of the lengths of the other two
  sides, then the triangle is acute.
• If c2 lt a2 b2, than ?ABC is acute.

10
Classifying a Triangle

• ?Classify the following triangles as acute,
  obtuse, or right.
• 85, 84, 13
• 6, 11, 14
• 7, 8, 9