The Pythagorean Theorem and its Converse

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Section 7-2

• The Pythagorean Theorem and its Converse

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GEOMETRY LESSON 7-2
(For help, go to the Skills Handbook, page 715.)
Square the lengths of the sides of each triangle.
What do you notice?
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Pythagorean Theorem

• In a right triangle, the sum of the squares of
  the lengths of the legs is equal to the square of
  the length of the hypotenuse.
• a b c

2
2
2
a and b can be used for either leg c has to be
the hypotenuse
c
a
b
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GEOMETRY LESSON 7-2
a2 b2 c2 Use the Pythagorean Theorem.
162 302 c2 Substitute 16 for a and 30 for b.
256 900 c2 Simplify. 1156 c2
34 c Take the square root.
The length of the hypotenuse is 34.
The lengths of the sides, 16, 30, and 34, form a
Pythagorean triple because they are whole
numbers that satisfy a2 b2 c2. Notice that
each length is twice the common Pythagorean
triple of 8, 15, and 17.
7-2
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Find X
x
3
4
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GEOMETRY LESSON 7-2
a2 b2 c2 Use the Pythagorean Theorem.
x2 102 122 Substitute x for a, 10 for b,
and 12 for c.
x2 100 144 Simplify.
x2 44 Subtract 100 from each side.
7-2
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GEOMETRY LESSON 7-2
a2 b2 c2 Use the Pythagorean
Theorem.
902 902 c2 Substitute 90 for a and
for b.
8100 8100 c2 Simplify. 16,200 c2
The distance to home plate from second base is
about 127 ft.
7-2
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Pythagorean Triple

• Set of nonzero whole numbers that satisfy the
  equation a b c
• 3,4,5
• 5,12,13
• 7,24,25

2
2
2
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Theorem 7-5Converse of the Pythagorean Theorem

• If the square of one side of a triangle is equal
  to the sum of the squares of the other two sides,
  then the triangle is a right triangle.

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GEOMETRY LESSON 7-2
7-2
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Theorem 7-6

• If the square of the length of the longest side
  of a triangle is greater than the sum of the
  squares of the lengths of the other two sides,
  the triangle is obtuse.
• If c²gta² b², the triangle is obtuse.

c
a
b
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Theorem 7-7

• 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, the
  triangle is acute.
• If c²lta² b², the triangle is acute

c
a
b
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GEOMETRY LESSON 7-2
Because c2 gt a2 b2, the triangle is obtuse.
Because c2 a2 b2, the triangle is a right
triangle.
7-2
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HOMEWORK

• Page 360
• 2-42e, 48-58e
• 7-2 Worksheet