Use the Pythagorean Theorem to solve problems Use Pythagorean Inequalities to classify triangles

1. Obj. 23 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. 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
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
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
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=
x 2 10=
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=
x 2 10=
2 2 2
4 (x 2) 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 2 2
4 (x 2) x+ − =xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
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+ − =xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 x 4x 4 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 2
16 x 4x 4 x+ − + =
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+ − + =
20 — 4x = 0
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+ − + =
20 — 4x = 0
20 = 4x
x = 5
2
6
x
x x-2
4

19. 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

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

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

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

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

25. 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 ,< +

26. 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

27. 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