The document discusses the Pythagorean theorem, which relates the sides of a right triangle. It states that the sum of the squares of the two shortest sides equals the square of the longest side (the hypotenuse). Examples are provided to demonstrate using the theorem to find missing sides of right triangles and to classify triangles as right, obtuse, or acute based on side lengths. Memorization of common Pythagorean triples like 3, 4, 5 is also encouraged.

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 TheoremPythagorean TheoremPythagorean TheoremPythagorean 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. that is
presumed to be 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: the hypotenuse is alwaysRemember: the hypotenuse is alwaysRemember: the hypotenuse is alwaysRemember: the hypotenuse is always cccc....
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: the hypotenuse is alwaysRemember: the hypotenuse is alwaysRemember: the hypotenuse is alwaysRemember: the hypotenuse is always cccc....
x
12
13
+ =2 2 2
12 13x

5. The Pythagorean Theorem allows us to find an unknown side
of a right triangle if we know the other two sides.
Remember: the hypotenuse is alwaysRemember: the hypotenuse is alwaysRemember: the hypotenuse is alwaysRemember: the hypotenuse is always cccc....
x
12
13
+ =2 2 2
12 13x
+ =
=
= =
2
2
144 169
25
25 5
x
x
x

6. 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
is an irrational number, like π, which goes on forever.
• If I ask for an exactexactexactexact answeransweransweranswer, I do notnotnotnot want a decimal – I
want you to leave it as a simplified radicalsimplified radicalsimplified radicalsimplified radical.

7. 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

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

9. 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 10x =
2
6
x
x x-2
4

10. 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 10x =
2 2 2
4 ( 2)x x+ − =
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
4 36 x+ =
2
40 x=
2 10x =
2 2 2
4 ( 2)x x+ − = xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 4 4x x 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
40 x=
2 10x =
2 2 2
4 ( 2)x x+ − = xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 4 4x x x+ − + =
20 – 4x = 0
20 = 4x
x = 5
2
6
x
x x-2
4

13. Pythagorean TriplePythagorean TriplePythagorean TriplePythagorean Triple
A set of nonzero whole numbers a, b, and c, such that
a2 + b2 = c2.
Memorize these!
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
××××2222 6, 8, 10 10, 24, 26 14, 48, 50 16, 30, 34
××××3333 9, 12, 15
x4x4x4x4 12, 16, 20
x5x5x5x5 15, 20, 25

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

15. Examples
Find the missing side of the right triangle.
1. 3, 4, ____
2. 9, ____, 15
3. ____, 12, 13
4. 8, 15, ____
5
12
5
17

16. Converse of the Pythagorean Theorem
If a2 + b2 = c2, then the triangle is a right triangle.
Another way to think of this is
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< +
2 2 2
.c a b= +

17. 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

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

19. Examples
Classify the following triangle measures as right, obtuse, or
acute.
1. 5, 7, 10
102 = 100, 52 + 72 = 74 ObtuseObtuseObtuseObtuse
2. 30, 16, 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