The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of using the theorem to find missing sides of right triangles and to classify triangles as right, obtuse, or acute based on side lengths. It also introduces Pythagorean triples, which are sets of integers that satisfy the Pythagorean theorem.

1. 4.11.1 The Pythagorean Theorem
The student is able to (I can):
• Use the Pythagorean Theorem to solve problems.
• Use Pythagorean inequalities to classify triangles.

2. The Pythagorean Theorem (a2 + b2 = c2)
states the relationship between the sides
of a right triangle. Although it was named
for Pythagoras (circa 500 B.C.), this
relationship was actually known to earlier
people, including the Babylonians,
Egyptians, and the Chinese.
A Babylonian
tablet from
1800 B.C.
listing sides of
right triangles.

3. The Pythagorean Theorem allows us to find
an unknown side of a right triangle if we
know the other two sides. Remember: theRemember: theRemember: theRemember: the
hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.
x
12
13

4. The Pythagorean Theorem allows us to find
an unknown side of a right triangle if we
know the other two sides. Remember: theRemember: theRemember: theRemember: the
hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.
x2 + 122 = 132
x
12
13

5. The Pythagorean Theorem allows us to find
an unknown side of a right triangle if we
know the other two sides. Remember: theRemember: theRemember: theRemember: the
hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.
x2 + 122 = 132
x2 + 144 = 169
x
12
13

6. The Pythagorean Theorem allows us to find
an unknown side of a right triangle if we
know the other two sides. Remember: theRemember: theRemember: theRemember: the
hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.
x2 + 122 = 132
x2 + 144 = 169
x2 = 25
x
12
13

7. The Pythagorean Theorem allows us to find
an unknown side of a right triangle if we
know the other two sides. Remember: theRemember: theRemember: theRemember: the
hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.
x2 + 122 = 132
x2 + 144 = 169
x2 = 25
x = 5
x
12
13

8. Square Roots
• When we are taking the square root of a
number, we will not always get a whole
number answer.
• If your answer is not a whole number,
then the number your calculator gives
you is a decimal approximationapproximationapproximationapproximation. This
number is like π, it goes on forever.
• If I ask for an exact answerexact answerexact answerexact answer, I do notnotnotnot
want a decimal — I want you to leave it
as a simplified radicalsimplified radicalsimplified radicalsimplified radical.

9. To simplify a radical (square root):
• Find all the prime factors of the number
• Group pairs of factors — these can be
pulled out of the radical
• Any factors that cannot be paired up
must stay inside the radical
Example: Simplify 24
24
2 12
2 6
2222 3333
=i2 2 3 2 6

10. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2
6
x
x x-2
4

11. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =2
6
x
x x-2
4

12. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
6
x
x x-2
4

13. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
2
6
x
x x-2
4

14. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
x 2 10=
2
6
x
x x-2
4

15. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
x 2 10=
2 2 2
4 (x 2) x+ − =
2
6
x
x x-2
4

16. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
x 2 10=
2 2 2
4 (x 2) x+ − =xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2
6
x
x x-2
4

17. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
x 2 10=
2 2 2
4 (x 2) x+ − =xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 x 4x 4 x+ − + =
2
6
x
x x-2
4

18. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
x 2 10=
2 2 2
4 (x 2) x+ − =xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 x 4x 4 x+ − + =
2
6
x
x x-2
4

19. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
x 2 10=
2 2 2
4 (x 2) x+ − =xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 x 4x 4 x+ − + =
20 — 4x = 0
2
6
x
x x-2
4

20. Examples Find the value of x. Reduce radicals to
simplest form.
1.
2.
2 2 2
2 6 x+ =
2
4 36 x+ =
2
40 x=
x 2 10=
2 2 2
4 (x 2) x+ − =xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 x 4x 4 x+ − + =
20 — 4x = 0
20 = 4x
x = 5
2
6
x
x x-2
4

21. Pythagorean
Triple
A set of nonzero whole numbers a, b, and c,
such that a2 + b2 = c2.
Memorize these!
Note: 3, 4, 5 is the onlyonlyonlyonly triple that
contains three consecutive numbers.
Pythagorean TriplesPythagorean TriplesPythagorean TriplesPythagorean Triples
BaseBaseBaseBase 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17
x2x2x2x2 6, 8, 10 10, 24, 26 14, 48, 50 16, 30, 34
x3x3x3x3 9, 12, 15
x4x4x4x4 12, 16, 20
x5x5x5x5 15, 20, 25

22. Examples Find the missing side of the right triangle.
1. 3, 4, ____
2. 9, ____, 15
3. ____, 12, 13
4. 8, 15, ____

23. Examples Find the missing side of the right triangle.
1. 3, 4, ____
2. 9, ____, 15
3. ____, 12, 13
4. 8, 15, ____
5555

24. Examples Find the missing side of the right triangle.
1. 3, 4, ____
2. 9, ____, 15
3. ____, 12, 13
4. 8, 15, ____
5555
12121212

25. Examples Find the missing side of the right triangle.
1. 3, 4, ____
2. 9, ____, 15
3. ____, 12, 13
4. 8, 15, ____
5555
12121212
5555

26. Examples Find the missing side of the right triangle.
1. 3, 4, ____
2. 9, ____, 15
3. ____, 12, 13
4. 8, 15, ____
5555
12121212
5555
17171717

27. Thm 5-7-1
Thm 5-7-2
Converse of the Pythagorean Theorem
If a2 + b2 = c2, then the triangle is a right
triangle.
Pythagorean Inequalities Theorem
If then the triangle is an
obtuseobtuseobtuseobtuse triangle.
If then the triangle is an
acuteacuteacuteacute triangle.
2 2 2
c a b ,> +
2 2 2
c a b ,< +

28. Classifying Triangles
right triangle
obtuse
triangle
acute
triangle
> +2 2 2
c a b < +2 2 2
c a b= +2 2 2
c a b
a
b
c
a
b
c
a
b
c

29. Examples Classify the following triangle measures as
right, obtuse, or acute.
1. 5, 7, 10
2. 16, 30, 34
3. 3.8, 4.1, 5.2

30. Examples Classify the following triangle measures as
right, obtuse, or acute.
1. 5, 7, 10
102 = 100, 52 + 72 = 74 ObtuseObtuseObtuseObtuse
2. 16, 30, 34
342 = 1156, 162 + 302 = 1156 RightRightRightRight
3. 3.8, 4.1, 5.2
5.22 = 27.04, 3.82 + 4.12 = 31.25 AcuteAcuteAcuteAcute