The Pythagorean Theorem

1
The Pythagorean Theorem

• A relationship between sides of a right triangle.

2
Vocabulary

• Theorem A statement that can be proven to be
  true.
• Hypotenuse The side opposite the right angle in
  a right triangle.
• Legs Sides of a right triangle that form the
  right angle.
• Converse A related statement in which the
  hypothesis and conclusion are interchanged.

3
The Pythagorean Theorem

• If a triangle is a right triangle, then the sum
  of the square of the lengths of the legs a and b
  equals the square of the hypotenuse c.
• a2 b2 c2

c
a
b
4
Use the Pythagorean Theorem

• Example 1 Given a12 and b5, find c.
• 122 52 c2
• 144 25 c2
• v(169) v(c2)
• 13 c

c
12
5
5
Use the Pythagorean Theorem

• Example 2 Given a3 and c5, find b.
• 32 b2 52
• 9 b2 25
• v(b2) v(16)
• b 4

5
3
b
6
Use the Pythagorean Theorem

• Example 3 A right triangle has a hypotenuse that
  is 1 foot longer than the longer leg. The shorter
  leg is 9 feet. Find the missing lengths.
• 92 x2 (x1)2
• 81 x2 x2 2x 1
• 81 2x 1
• 80 2x
• 40 x
• The longer leg is 40 feet and the hypotenuse is
  41 feet.

x1
9
x
7
Converse of the Pythagorean Theorem

• If a triangle has side lengths a, b, and c such
  that a2 b2 c2, then the triangle is a right
  triangle.

8
Determine Right Triangles

• Example 4 Determine whether the lengths 20, 21,
  29 are sides of a right triangle.
• 202 212 292
• 400 441 841
• 841 841
• In a right triangle, the hypotenuse is always
  the longest side.

9
Use the Pythagorean Converse

• Example 5 Erica is setting out the boundary of a
  tool shed that she is going to build. The
  building will be 8 feet long and 6 feet wide. The
  diagonal of the boundary measures 11 feet. How
  does Erica know that the corners of her boundary
  are not 90?
• 82 62 112
• 64 36 121
• 100 ? 121

10
Skill Check

• Find the missing length of the right triangle.
• a 7, b 24
• a 5, c 13
• b 15, c 17
• Explain how you can use the converse of the
  Pythagorean Theorem to tell whether three given
  lengths can be sides of a right triangle.
• (Solutions on next slide.)

11
Skill Check Solutions

• Find the missing length of the right triangle.
• a 7, b 24, c 25
• a 5, c 13, b 12
• b 15, c 17, a 8
• Explain how you can use the converse of the
  Pythagorean Theorem to tell whether three given
  lengths can be sides of a right triangle.
• If a2 b2 c2, then the sides a, b, and c form
  a right triangle.