4to Año - Guia Nº 6 - Triángulos II - Líneas Notables

I BIM GEOMETRA 4TO. AO

EJERCICIOS DE APLICACIN NIVEL 1

1. En la figura; calcular x

a) 108

b) 54

c) 72

d) 36

e) 44

2. Calcular x

a) 55

b) 60

c) 45

d) 40

e) 10 3. Calcular x

a) 100

b) 120

c) 130

d) 150

e) 170

4. Calcular x a) 100

b) 80

c) 125

d) 150

e) 250

5. Calcular x

a) 85b) 75c) 70d) 65e) 60 NIVEL 2

6. En la figura, calcule x

a) 10

b) 20

c) 65

d) 35

e) 45

7. En la figura, calcule x a) 35

b) 30

c) 15

d) 10

e) 20

8. En la figura ; Calcule x

a) 125

b) 155

c) 115

d) 100

e) 20

9. Del grfico, calcule x ;

a) 52

b) 48

c) 44

d) 42

e) 40

10. Del grfico, calcular x

a) 110

b) 90

c) 70

d) 20

e) 10

NIVEL 3

11. En el grfico, AB = BC Calcule x

a) 45

b) 120

c) 60

d) 70

e) 37

12. Determine x, Si : son madiatrices de .a) 30b) 15c) 20d) 36e) 4513. Calcular x a) 90b) 100c) 120d) 130e) N.A.14. Calcular x; si es un valor entero mximo. son bisectrices exteriores de los ngulos B y C; respectivamente.

a) 3b) 7c) 8d) 9e) 1015. Segn el grfico, calcular el valor x

a) 110b) 120c) 130d) 150e) 95

TAREA DOMICILIARIA N 6 NIVEL 11. Calcular x

a) 100

b) 80

c) 40

d) 20

e) 10


a) 60

b) 45

c) 35

d) 75

e) 55


a) 60b) 25

c) 50

d) 40

e) 20

4. Calcular x

a) 15

b) 30

c) 45

d) 60

e) 75

5. Calcular x;

a) 30b) 35c) 60d) 75e) 45 NIVEL 2 6. En el grfico. Calcular x

a) 15

b) 20

c) 25

d) 30

e) 35

7. De la figura; 3( = 5(Calcular x

a) 25

b) 15

c) 30

d) 20

e) 35

8. Calcular x ;

a) 135

b) 115

c) 112,5

d) 52,5

e) 22,5

9. De la figura : 5(mAED) = 6(mADC)

y mBAD = 70; Calcular la mCAD

a) 44

b) 24c) 14d) 15e) 10

10. Segn el grfico mostrado: Calcular : ( +(

a) 100

b) 150

c) 90

d) 180

e) 270

NIVEL 311. Del grfico, Calcular x

a) 15

b) 8

c) 10

d) 5

e) 25

12. De la figura, calcular x; en funcin de (

a) 90-

b) 45+

c) 45-

d) 90+

e) 45-

13. De la figura, calcular : x + y + z

a) 180

b) 360

c) 300

d) 270

e) 100

14. Calcular x

a) 27

b) 45

c) 30

d) 36

e) 18

15. En el grfico, calcular x

a) 150

b) 110

c) 120

d) 100

e) 135

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Naci : Arededor del ao 640 a.C. en Mileto, Asia Menor (Ahora Turqua)

Falleci : Alrededor 560 a.C. en Mileto, Asia Menor (Ahora Turqua)

Thales era un hombre esencialmente prctico: comerciante, hbil en ingeniera, astrnomo, gemetra, estadista. Se le incluye por tradicin entre los Siete Sabios.

Como comerciante se cuenta de l que un ao, previniendo una gran produccin de aceitunas, monopolizo todos los lugares para hacer el aceite; con lo cual obtuvo esplndida ganancia. Como lo que ahora llamaramos ingeniero, estuvo dirigiendo obras hidrulicas y se dice que desvi el curso del ro Halis mediante la construccin de diques.

Como astrnomo fue ms clebre, predijo el eclipse total de sol visible en Asia Menor, como asimismo se cree que descubri la constelacin de la Osa Menor y que consideraba a la Luna 700 veces menor que el sol. Tambin se cree que conoci la carrera del sol de un trpico a otro. Explic los eclipses de sol y luna. Finalmente crea que el ao tena 365 aos.

A Thales se le atribuyen 5 teoremas de la geometra elemental.

1. Los ngulos de la base de un tringulo issceles son iguales.

2. Un crculo es bisectado por algn dimetro.

3. Los ngulos entre dos lneas rectas que se cortan son iguales.

4. Dos tringulos son congruentes si ellos tiene dos ngulos y un lado igual.

5. Todo ngulo inscrito en una semicircunferencia es recto.

Thales busca el fundamento natural de las cosas y cree, al respecto, que el principio originario, la sustancia primordial de todas las cosas, es el agua.

Pensaba as mismo que el agua llenaba todo el espacio. Se imaginaba a la Tierra como un gran disco flotando sobre las aguas, sobre la cual existira una burbuja hemisfrica de aire, nuestra atmsfera sumergida en la masa lquida.

La superficie convexa de la burbuja sera nuestro cielo y los astros segn expresin de Thales Navegaran por las aguas de arriba.

Escribi un libro de navegacin y se deca que uso la constelacin de la Osa Menor que l haba definido como una caracterstica importante de la navegacin.

Se cre que Thales pudo haber sido el maestro de Anaximandro y que fue el primer filsofo natural de la Escuela Milesiana.

Su busto se exhibe en el museo del capitolio en Roma, pero no es el contemporneo de Thales.

TEOREMA DE THALES DE MILETO

Teorema de Thales, relacin bsica para obtener las propiedades fundamentales de la semejanza de tringulos.

Segn este teorema, una familia de rectas paralelas r1 , r2 , r3 , que cortan a dos retas concurrentes, s y t, determinan en ellas segmentos proporcionales:

NIVEL: SECUNDARIASEMANA N 6CUARTO AO

TRINGULOS II LNEAS NOTABLES

(

(

x

(

(

72

B

x

80

a

a

b

b

C

A

x

100

60

(

60

(

100

x

(

(

(

(

170

(

(

(

x

(

80

60

x

(

(

(

(

2(

70

(

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(

30

70

(

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x

(

B

A

(+10

x

D

C

(

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(

40

(

(

(

x

(

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20

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40

x

(+(

(

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x

C

A

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B

(

2(

L1

B

L2

x

75

P

A

Q

C

x

(

(

(

(

x

P

B

7

3

x

C

A

2(

2(

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x

(

150

2(

x

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50

(

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B

50

x

80

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C

A

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x

130

P

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C

E

B

Q

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x

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30

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120

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b

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PAGE 130Lic. Joel Dik MARCOS ESPINOZA

Matemtica y Fsica

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4to Año - Guia Nº 6 - Triángulos II - Líneas Notables

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