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Notes - Geometry
7.1 and 7.2: Pyth agorean Th
Background: The Pythagorea
mathematics. The Scarecrow
famous Greek mathematician
VERY long time ago). This the
sides of a right triangle.
best known theorem of all times in
movie "The Wizard of QZ". A very
around 500 B.C. (yes - that is a
lformation about the lengths of the
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The Pythagorean Theorem:
T",
D--...
-"'%'rtt-: ~\a.+
~ +!w.. Sh~ ~ \-)9..
l
~
\
Th_
6~
I
S%\.LC>.l'Q .
-'U!_
Example: Find the length of the unknown side.
~ ')..
lo
3\0+ \0'-\
\1:)0
-=- c...':t
=
-=-
~~
8
(., 1­
Example: Identify the unknown side as a leg or hypotenuse. Then, find the unknown side
length of the right triangle.
'-\ 1.. -\-
J2 +- X
1.
5
3
x
Geo Notes 7] 72 2007 2008
S L.­
c; t- 'Y-.1-. =- d,ts
7...
=
x.t- =- \~
~
\4U
2.
\ \S)
4
x
6
\0 "L =- >< '2­
+ 3~
5'2.. -::
=..
X. 1­
X
i­
Example: A 16 foot ladder rests against the side of a house, and the base of the ladder is 4
feet away_ Approximately how high above the ground is the top of the ladder? (Draw a
picture!)
a) 240 feet
c) 16.5 feet
b 20 feet
d) 15.5 eet
X
..,
X''1.. -=- \ LP {.-
1.-\ L.. +-
'J.
'1.. ... "\
'"
-0
4b
\lJ:,+)('"' ~ d5~
X=-\S,4q
4
Example: Randy made a ramp for his dog to get into his truck. The ramp is 6 feet long and
the bed of the truck is 3 feet above the ground. Approximate how far from the back of the
truck does the ramp touch the ground?
3 'l--\- 'X '1 ~ \p 2­
b) 4 ft
d) 6.7 ft
@ift
=.2ft )
31.P
~ t-')( L ~
x"'"2.-:::..
X -:::
Question?: The Pythagorean Theorem is only true for what type of triangle?
dl
S,~t>
--D\ ~bt
Example: Find the area of the isosceles triangle with side lengths 20 meters, 13 meters and
13 meters.
\\:) "L \- y., '1. -::..\3 '­
\ ~CJ +-- 'K
"L :::.. \
\.!I C\
x. L.:::
\0 c;,
X -:::
~C\
~()
Pythagorean Triples: A Pythagorean Triple is a set of three positive integers a,b and c that
satisfy the equation
a2 + b2 = c 2 .
Common Pythagorean Triples and Some of their Multiples
3,~\S
5 \ \~ \ \3
~\lS"\\1
., \dl..\q:l$
n~\~\31.4
\'-\-\
Lt~, S"D
.ct \ \:J.. \ \S \S \ 3lD\5C1 d4 14S,S\ d-.\ ;72,/5
:JJ , L\\) ~O S'D\ \d.D \Be ~t> \ \SO, 111) {()\~4t\ ;}SO
ltJ \ ~ \ \0
\D \ d.,"", \ ~\..o
.......
~,
It will help be very helpful if you memorize the basic Pythagorean triples, the ones in
the top row, for standardized tests - especially the SAT and SOL!
Geo Notes 71 72 2007 2008
2
The Converse of the Pythagorean Theorem:
~
VDL :'~1J>AA1., ~ fu
'S\~
~ e>:- ~ \~ ~\n\)
~u..OS\9.b
~ ~
~ :!:'<\.o....
--:r 'b
C-.')..
=
.6
Q:~
\S
k~ . ~ ~ ~t
~~'S
.J".'%'nt
\5. 0 -
'0 ':l \ ~
...5\
0
13
f::,. F\:?,<,­
T-
C1-.
~ 0W'sn ~ "\=h9-­
~ -\-~ ~ d S\ 6.2.D I
6
~,
~~t
c
a
4-,----b---"'"
Tell whether a triangle with the given side lengths is a right triangle? (Do you remember how
to tell whether 3 numbers even represent the sides of a triangle? Do you remember how to
determine the possible range of the length of the third side of a triangle??)
'--"
~
,-\L.+~t3)i.. D 6~
1. 4,413, and 8
t~+ ~~
D~4
2. 10,11and14
No
3. 5,6 and
lD"L+\\LD \\..1.;1.­
\D(J+
1...\ D
5'2 -\- \..s>L
\C\lo
~
- ~J"2.
.).S~ 3~ D
~~\
lDL\
.J6i
\
\
You can use the converse of the Pythagorean Theorem to classify a triangle acute,obtuse
OR right! Very useful!!!
If
\) L
If G
L
'=
~<.. ~"c 1.­
'/0.-"1- -\--'0 t.-
, then the triangle is 0-- J\\~"'\:- 6,
, then the triangle is
, then the triangle is
Geo Notes 71 72 2007 2008
lJ~~
~~
0......
.
3
f\
\~
'S~~\:.
~ ~
~~
L.
3S\C\ \
•
,
Examples: Decide if the segment lengths form a triangle. If so, would the triangle be acute,
right or obtuse?
4. 10,11 and 14
~
5. 10,15 and
sm
-.5""\,,~~t
\ ~+ \S l.. a~5"\3) '­
\\)0 ~ ~ d.-S
3 ~S
O~
d~d. ~ :c>"1;l ~~3') 'S"l~+ Citb D \5'-\6
6. 24, 30 and" 6.J43
\L\\\p
/
\'5
--=- '32-5
N"S't
0-
~
~ cl\:)
:J5
<... \S"l..\~
8. Which side lengths do not for a r"
a) 5,12,13
7. 15,20,36
'0 3.;;lS
b) 10,24,28
.ange.
I?
-:>
S, \'2.. ,\::>
c) 15,36,39
d) 50,120,130
-\OO+-S7~LJl6l-\
S \ \1- \ \ ~
9. What type of triangle has side lengths of 4, 7and 9?
~
(0)
obtuse scalene
Geo Notes 7J 72 2007 2008
b) right scalene
d) None of the above
4