Logo Studenta

Geometria Plana e Geometria Espacial

¡Este material tiene más páginas!

Vista previa del material en texto

� �
� � � � � � � � � � � 	 � 
 � � � 
 � � � � � � � � � �
�
� � � � � � � � � � � 	 	 � 
 � � � 
 � � � 	 � � � 	 � � � � � � � � � � � � � � � � � 
 � 	 � � � � � � 	 � � � � � � � � � � � � � � 
 � � � � � �
� � � � � � 	 � � � � � � � � � � � � � � 	 � � � � � � ff
fi fl ffi � ! " # $ %
& '
ff
�
� � � � � � � � � � � � � � � � ( � ) � � � � � � ( � � � � � � * � 	 � � � � � � 	 	 ) � + , � - - � 
 � �
&
+ � . - � /
+
'
ff 0 � � � � � � � � � � � � � ( � � � � � � � 1 � , - � 
 � �
&
. � - - � � � 	 � � 	 � � � � ( � � � � � � ff 2 � � � � � � � � � � � � � � � �
� � 	 � � � � � � � � ( � � � � � 	 � � � � � � � � � 
 � � � ( � � � � � 3 � - - � 4 � � � � � � � � � � � � � � � � 	 � � � � � 	 � � 	 � � � � � � � �
( � � � � 	 ) � � � � � 	 	 � � � � 	 
 � � � ( � � � � � � � � � � � � � � � � � /
3
'
ff 0 � � � � * � � � � � � � � ( � � � � � � � 3 - � � 
 � � + - � � ff
�
� � � � � � � � � � � � � * � � � /
�
� � � � � 	 � � * � � � 	 	 ) �
� � � � 	 	 � � � � 	 
 � � � � � � � � � � 
 � 	 � � � � � � ( � � � ( � � � � 5 6 � - - � 4 � � � � � � /
,
'
ff
�
� � � � � 	 � 4 � � � 7 � � � * � 	 ) � � � � � 	 	 � � � � 	 
 � � � � � 8 � 	 � � � � � � � � � � � � � � 
 � � � � � � � � � � ( � � � � � � , � - -
� 
 � � + � 9 . � /
fi fl ffi � ! " # $ %
.
'
ff 0 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 3 - � + - � ff : � � � � � � � � � � � � � 	 	 � � � � � � � � ff
6
'
ff 0 � � � � � � � ; � � � � � � � � � � � � � � � � � � � + - � � � � � � � � ff
�
� � � � � � � � � � � 	 	 � � � � � � � ; � /
9
'
ff < � � � � � � � � � ; � � � � � � � � � � � � � � � 
 � � � � � � � + 9 � - - � 4 � � � � � � � � � � � � 	 � � � 	 
 � = � 	 � � � � � � � � 	 � �
&
. � � � � � � � � ff
�
� � � � � 	 
 � = � 	 � � � � � � 	 � � � 	 /
� >
fi fl ffi � ! " # $
1
'
ff ? � � ( � ) � � � � � � � � � � � � � � � � � � � � � � � � 
 � � � � 
 � � � � � � � ( � � � � � � � � � �
&
. � , � � � � � � � 
 � � � � � � �
 � � 1 � . � � � � � � � ( � � � ff
�
� � � � � � � � � � � 	 	 � � � ( � ) � /
5
'
ff 0 � 
 � � � = � � � � � � 
 � � 	 � � � � � � * � � 	 
 � 	 	 � � � � � � 	 
 � � 7 
 8 � � � � � � � � � � � � � � 	 � � � � � 	 @ � 	 � �
� � ( � � � � � � � A � ff B � � � � � � � � � � � � � � � 	 	 � 
 � � � = � � � � � � 
 � � 	 � � � ff
? � � � � � � � � � � � C � ( � � � � � � � � � 
 � � � 	 � � � � � � � � � � � � � � � D E � � � � � � � F � � � � �
� G
fi fl ffi � ! " # $ %
&
-
'
ff 0 � 
 � � � = � � � � � � � � � � � � � * � � 	 
 � 	 	 � � � � � � 	 
 � � 7 
 8 � � � � � � � � � � � � � � 	 � � � � � 	 @ � 	 � � � � ( � � �
� � � � A � ff : � � � � � � � � � � � � � 	 	 � 
 � � � = � � � � � � � � � � ff
& & '
ff 0 � 
 � � � = � � � � � � � � � � � � � � � � � � � � � � � 	 � � � � � 	 @ � 	 � � � � ( � � � � � � � A � ff
�
� � � � � � � � � � � 	 	 �
 � � � = � � � � � � � � � � � � /
fi fl ffi � ! " # $ %
&
+
'
ff
�
� � � � � � � � � � � � � � � � C � ( � � � � � � C � ( � � � � � * � ; � 
 � � � � � 	 � � � � �
&
3 � � � � � � � 	 � � � � � � 	 � � � �
. � � /
� H
&
3
'
ff : � � � � � � � I 	 � � � 
 � � � = � � � � � � � � � � � � � � � 8 � I 	 � � � � ( � � � � � � � A � � � � � 	 � � 	 � � � � � 	 @ � 	
� 
 � � A � � � � � 	 ff : � � � � � � � � � � � � � 	 	 � 
 � � � = � � � � � � � � � � ff
fi fl ffi � ! " # %
&
,
'
ff < � � � � � � � � 	 � � * � � � � � � � � � � � � � � � � � � � � 	
&
- - � � � � � � � � � ; � 	 � � � � � � � � � � � � � � � � � � C � ( � � �
� � J � � � � � � � � � � � � � + - � � ff
�
� � � � � 	 � 4 � � 
 � 
 � � � � � � � � � � � 	 	 � � � � 	 
 � � � � � � � � � 	 	 � 	 � � � � � � � � � ; � 	 /
K
 # L M ffi ffi L N # O $ ffi P $ Q ffi N " N P $ N ffi ffi P Q ffi N " N P
N # R # � Q P N # S # � ffi $ $ ffi $ T P N # $ %
U V
fi fl ffi � ! " # %
&
.
'
ff 0 � � 
 � � � � � � � � � � � � � � � � � � � � � � � � ( � � � � � � � A � ff 2 � � 	 � � � � � � 	 � 	 � ) � � � � � � � � � 	 � � � � ( � � � ff
: � � � � � � � � � � � � � 	 	 � 
 � � � � � � � � � � � ff
W X Y Z
0 � � � � � � � � � � � � � � � � � � � 	 � � � � � 	 @ � 	 � � � � ( � � � � � � � A � ff : � � � � � � � � � � � � � 	 	 � � � � � � � � ff
fi fl ffi � ! " # %
&
9
'
ff ? � � ( � � � 	 � ( � � � � � � � 	 � � 	 � � � � � � � � � � � � � � � [ � � � � � � � � � � � � \ � � � � � � � , � � � � � � � 	 � � ( �
? � ] � : � B � � � * � � � � � � � � � . � � ff : � � � � � � � � � � � � � 	 	 � � � 	 � � ( � ff
U �
^ _ ` a b ffi c ffi � Q " L ffi P d � ffi P N # T # $ P L e # � ffi S � ffi $ ffi L c P N # S ffi T P R " e f � P a
fi fl ffi � ! " # $ %
&
5
'
ff \ � � � � � � � � � � � � ? � ] � : � B � � � � � � 
 � 7 � � � � * � 	 � � 	 � 	 � � � � � 3 - � � � +
&
� � ff 2 � � � � � � � � � �
� � � � � � � � 	 	 � � � � 
 � 7 � � �
&
6 � � � � � � � � � � � � � � � � � � � � � � 
 � 7 � � ff
+ -
'
ff D � � � � � � � 8 � � � � � � � � � � � 	 � � � � � � 	 � � � � � � � � � � � 
 � � � � ( � � � � � � � � � � � � � � � � � � � � � 	 � 	 � � � � 	
� � � � � � � � 	 � � � � ( � � � ff
U U
fi fl ffi � ! " # $ %
+
& '
ff g � � � � � 
 � � � � � � � � � � � � ( � � � � � � 
 � � � � � � � �
&
- � � � � � � � ff
�
� � � � � � � � � � � ( � � � � � � 
 � � � � � /
+ +
'
ff
�
� � � � � � � � � � � � � ( � ) � 	 � � � � � � � � � 	 � � � � � � I 	 � � � � ? � ] � : � B � � � � � � � � � � � � �
&
6 � � � �
 � � 
 � � � � � /
+ 3
'
: � � � � � � � 	 � � � � 	 � � 	 	 � ( � � � � � 	 � � ( � � � 	 ( � � � � � � � � � 	
U h
? i � � � � � � � � � 	 
 � � � � � � � � � � � � � � � � � � � � � � 
 � � � = ) � � � i � � � � � � � � 
 � � � � � � � � � � � � 	 � � � � � � 	
� 
 � � 
 � � � � � 	 
 � � � � � 	 � � � � � � � � * � � � 	 � 	 
 � � � � � 	 � 	 	 � � � � � � � � � � � = ) � � � � � � � 	 	 � 	 � � � � � � � � 	 ff
? 	 	 � � � � 	 � � � � � � � � 	 � 	 
 � � � � � � � � � � � � � 	 � � � � � � � 	 � � � � � � � � 	 
 � � � � � � � � � = ) � � � 	 8 � � � � � 	 � � 	 � � 	
� � * � � � 	 ff
j � " $ Q P $
< � � 	 � � � � � 	 E � � � � ( � � � � � � � � � � � � � � � � � � � 
 � � � � � � 	 
 � � � � 	 � � � � � � � � 	 � � 	 � 	 	 � 	 � � � � � � � 
 � � � � 	
 � � � � � � � 	 ff
�
� � � � � k l m n o l m p q r s � � 	 � � � 	 � � 	 � � � � � � � 	 � � 	 
 � � 	 � � 	 
 � � � � 	 � � � � � � 	 � � � � � 
 � � � 	 ff
\ t � � � � � � � � � 
 � � 	 � � � � � � � � � � � � ( � � � � � 
 � � � � � � � � � � � � � � 	 � � � � 	 � 
 � � � � � � � � � ff
u v w x y z { z x |
B � � � � � � 	 � � * � � � 	 � � � � 	 � � � 
 � � � � � 	 � � 
 � � 	 � � � � � 
 � � 
 � � 	 � � 	 � � � � 	 � � � � � � � � � � � 
 � � � � � � � �
� � � � � � 	 � � � 
 � 	 	 � � � � � � � 	 � 	 � � � � 	 	 � � � � 
 � � � � � � � ( � � � � 	 ff } 	 	 � 	 
 � � 	 � � 	 � � � � � � � � � � � �
� 	 
 � � � � � � � 
 � � � � � � � 
 
 
 � � � ff
\ 8 � � � � � � � 
 � � � � � � � 
 
 
 � � � � � � � � 
 � � �
U ~
\ 8 � � � � � � � � � � � � � � � � � ( � � � k � � � � � � 	 � � � � 	 � 
 � � � 	 � � � � � � � � � � � 	 � * � �
U 
€
# L $ " N ffi � ffi f Q P � ffi e "  # S T P L P T " Q " c P N P S # � f Q P f � ‚ P
$ f P ‚ ffi ƒ $ ffi Q „ f " L P $ `
K
R ffi M P N P ffi f Q S # L c # j R # � P N ffi $ $ ffi
S T P L # a
€
M P Q P Q # $ N ffi # L ffi P # $ … T " N # R # � Q P N # S ffi T P
� ffi f L "  # N ffi c # N # $ # $ $ ffi e Q ffi L c # $ N ffi � ffi c P „ f ffi c † Q f Q P
ffi fl c � ffi Q " N P N ffi ffi Q j ffi P # f c � P L f Q S # L c # „ f P T „ f ffi � N P
� ffi e "  # a
‡ ˆ
# T f Q ffi N ffi f Q # L ffi ‰ " e f P T P f Q c ffi � Ł # N # S � # N f c # N P d � ffi P N P ‹ P $ ffi S ffi T P P T c f � P
K
# f $ ffiΠP %
U �
��
�
�
? � 	 � � � � � � � 	 
 � = � � � � � � * � � � � � � � � � � 	 � 	 
 � � � � 	 � � � 	 
 � = � � � �
� 	 � ) � � � � � � � 7 � � � 	 � � � � � � 	 � � � � 	 � C � � � � � � � � � � � � � � � � � � � � � � �
 � � � � � � A � � ; � � � � � � � � � � � ff
�
fi fl ffi � ! " # $ %
& '
: � � � � � � � 8 � � � � � � � 	 	 � ( � � � � � 	 	 E � � � � 	 �
+
'
0 � � � � � � � �  � � � � � � 
 � 
 � � � � �
&
+ � � � � 
 � � � � � � � � � � � � 1 � � � � � � C � � � � � ff B � � � � � � � � 	 � �
� � 
 � � � � � � � � � � � � � � � � � � 	 Ž
&
� � 
&
� � 
'
ff
3
'
0 � } � ( � � ; � � � � � 	 � � 
 � � * � � � � � � � � � 	 � 
 � � � Ž 
 � � � � � � � � � � � � � � � �
'
� � � � � � � � � � � � � � � � � � �
� � � � � � � � 
 � � C � � � � � � ( � � � � � � � � � 	 � � � � � 	 @ � 	 � � � � � � � � 	 � � � � ( � � � � � � � A � ff 2 � � � � � � I 	 � � � � � �
&
�  � � � � � � � � � � ( � 	 � � I 	 � � 
 � � A � � � � � � � � � � 5 	 � � � 	 � � � � � � � � � � � � � � � � � � � � � � � � � 	 	 � � � 	 	 � � ) �
( � 	 � � 	 
 � � � � � 7 � � � 	 	 � 	 � 
 � � � ff
U >
,
'
? � � 	 � � � � � � 
 � � C � � � � � � � � � � � � � � � � � � � � � 3 � � ff 2 � � � � � � I 	 � � � � � 
 � � C � � � � � � � � � � � � �
� �
&
- � � � � � � � � � � � � 8 � � � � � � � 	 	 � 
 � � C � � � � ff
.
'
: � � � � � � � � � � � 8 � � � � � � � � � � � 	 � � � � 
 � � � � � 	 � � � 
 � � � � � � 	 �
Ž
& '
� � � � � � � � � � � � � � 	 � � ( � � � � 3 �
&
� � � � � � � � � � �
& &
� 6 � � ‘
Ž +
'
� � � � � � � � � � � � � � 	 � � ( � � � � 3 �
&
� � � � � � � � � � �
&
6 � 6 � � ff
\ 	 
 � � = � 	 � � � � 	 	 � � � � � � � 	 ) � ’ “ - � 9 - � ’ “
&
�
&
- � � � 	 
 � � � � 8 � � � � � � � 
 � � � � 	 � � � � 	 Ž
& '
� Ž +
'
ff
: � � � � � � � 8 � � � � � � � � � � � � � � � 
 � � � � � ‘
�
� � � � � 	 � � � 	 � � � � � � ( � � 	 � 
 � � 	 � � � � � � � ; � � 
 � � = � 
 � � � � � � � 	 � � � � � � /
6
'
0 � 	 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ž � � � � � � � �
'
� � � � � � � � 
 � � � � � 	 � � � I � 	 � � � � �
� � � � � � � � ( � � � ff B � � � � � � � � � 8 � � � � � � � 	 	 � 	 � � � � 	 � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � - - � � � � �
� � � � � � � � � 	 � � � � � � � 1 � - - � ff
9
'
: � � � � � � � 8 � � � � � � � � � � � 
 E 	 � � � � � ( � 	 � 	 � � � � � � � � � � � � � � �
&
3 � . - � ff
1
'
< � � � 8 � � � � � � 7 � � � � 
 � � * � � � � � � � � � ( � = ) � � � � � � � 	 	 � � � � � � � 	 � � � � � � � � � � � � � � 	 � = ) � � � � 
 � 7 � � � � �
� � � � � � � � � � ( � � � � � � � A � � � � � 3 + . � - - � � � � � 	 � � � � � 
 � � � � � � � ff : � � 	 � � � � � � � � � � � � � � � � 
 � � � � � � � � �
� 8 � � � � � � � � 	 � � 8 � = ) � � � � � 	 	 � � � � 
 � � � � � � � 	 � � � = ) � � � 	 � � � � � � � /