Post on 18-Oct-2015
PAGE 360Razonamiento Matemtico Ronald Carhuancho
Geometra: Parte de la matemtica que trata de las propiedades y la medida de la extensin.
Punto: Limite mnimo de la extensin que se considera sin longitud, latitud ni profundidad.
Lnea: Esta formado por la sucesin continua de puntos con una sola dimensin que es la longitud.
Lnea recta:
Lnea curva:
Lnea quebrada:
Lnea mixta:
Lnea recta: sucesin contina de puntos que se desplaza hacia ambos extremos en forma ilimitada.
Semirecta: Parte de la recta que carece de punto de origen.
Rayo: Parte de la recta que posee punto de origen.
Segmento de Recta: Porcin de recta comprendido entre dos puntos que son los extremos.
Plano: Superficie imaginaria ilimitada, es engendrada por una lnea recta cuando se desplaza paralelamente a su posicin original.
Figura Geomtrica: Es un conjunto de puntos sistemas de lneas y superficies que reciben el nombre de figuras geomtricas.
Significado de los trminos matemticos:
Axioma: Es una proposicin evidente por si misma y que no necesita demostracin:
Teorema: Proposicin que mediante un razonamiento se hace evidente y consta de dos partes; hiptesis y tesis.
Corolario: Es una consecuencia de uno o varios teoremas.
Postulado: Proposicin que sin ser evidente se admite su certeza por no ser posible demostrarla.
Lema: Es un teorema preliminar que sirve de base para demostrar un teorema principal.
Escolio: Es una advertencia o anotacin que se hace para aclarar, ampliar o restringir proposiciones anteriores.
Proposicin: Es el enunciado de una hiptesis suposicin y conclusin.
Hiptesis: Punto de partida de una demostracin lgica a partir del cual se propone alcanzar la solucin.
Problema: Es una proposicin que se hace con el objeto de aclararlo resolverlo.
Operaciones con Segmentos:
Suma:
*
*
*
Resta:
*
*
*
Mximo nmero de puntos de corte
* Para puntos secantes
* Para circunferencias secantes
* Para polgonos secantes
1 Sobre una recta se ubican ordenadamente los puntos A, B, C y D. Si ,. Calcular la longitud de .
a) 4m
b) 8m
c) 9m
d) 5m
e) 3m
Resolucin:
*
Entonces:
Luego:
EMBED Equation.DSMT4 Rpta.
2 Sobre una lnea recta se ubican los puntos consecutivos A, B, C, D y E, donde:
; y , Calcular la longitud de .a) 2m
b) 4m
c) 15m
d) 3m
e) 17m
Resolucin:
*
Luego:
EMBED Equation.DSMT4 Rpta.3 Sobre una lnea recta se ubican ordenadamente los puntos A, B, C, D y E, si: y adems , calcular: .
a) 10m
b) 30m
c) 50m
d) 20m
e) 40m
Resolucin:
*
, entonces:
..( I )
Del dato:
Reemplazando ( II ) en ( I )
EMBED Equation.DSMT4 Rpta.
4 Sobre una recta se tienen los puntos consecutivos A, B, C, D, E, F y G cumplindose que:
AD + BD CD + CG + DG EG = 14, edemas se cumple que: . Hallar AG.Resolucin:
( I )
.( II )
Agrupando convenientemente en ( II )
( III )
Reemplazando ( I ) en ( III )
EMBED Equation.DSMT4 Rpta.5 En una recta se toman los puntos consecutivos L, I, M, O, N tal que M es el punto medio de . A que es igual:
Resolucin:
Sea:
Reemplazando
EMBED Equation.DSMT4 Rpta.
NGULO
Es la figura formada por dos rayos divergentes que tienen un extremo comn denominado vrtice.
Notacin: ; ,
Bisectriz: Rayo que divide al ngulo en dos ngulos congruentes.
Clasificacin:
Los ngulos se clasifican segn su magnitud, segn sus caractersticas y segn su posicin de sus lados.
I. Segn su Magnitud:
1: ngulo Nulo:
2: ngulo Convexo:
3. ngulo llano:
4. ngulo Cncavo:
5. ngulo de una vuelta:
II. Segn sus caractersticas
a) ngulos Complementarios
b) ngulos Suplementarios
III. Segn Posicin de sus lados
a) ngulos adyacentes suplementarios
b) ngulos Consecutivos
c) ngulos opuestos por el vrtice
ngulos de lados paralelos
a)
b)
c)
ngulos de lados perpendiculares
1) Dos ngulos agudos
2) Dos ngulos obtusos
3) ngulos: agudo y obtuso
ngulos formados por dos rectas paralelas y una recta secante
ngulos internos:
ngulos externos:
ngulos alternos internos:
ngulos alternos externos:
ngulos conjugados internos:
ngulos conjugados externos:
ngulos correspondientes:
Propiedades entre rectas paralelas:
1. Si:
2. Bisectrices de un par lineal
PROBLEMAS RESUELTOS
1 Dados dos ngulos consecutivos: AOB, BOC y COD, se cumple que , Calcular la medida del ngulo formado por las bisectrices de los ngulos AOB y COD.
Resolucin:
De la grfica se observa:
EMBED Equation.DSMT4 Rpta.
2 Se tienen tres ngulos consecutivos AOB, BOC y COD de tal manera que las bisectrices de los ngulos AOB y COD sean perpendiculares, donde el ngulo BOD mide 80. Calcular la medida del ngulo AOC.
a) 150
b) 100
c) 90
d) 60
e) 80Resolucin:
De la grafica:
De la grafica:
Pero:
( II ) en ( I )
EMBED Equation.DSMT4 Rpta.
3 En la grafica si //, Calcular la medida del ngulo x.
a) 10
b) 30
c) 50
d) 20
e) 40
Resolucin:
Por las propiedades entre dos rectas paralelas.
Por propiedad:
. ( I )
Cuadriltero cncavo
. ( II )
( I ) en ( II )
EMBED Equation.DSMT4 Rpta.
4 En la grafica mostrada calcular el valor del ngulo x, si
a) 40
b) 50
c) 70
d) 60
e) 80
Revolucin:
* Si es el complemento de x
Entonces el triangulo sombreado es equiltero:
De donde:
EMBED Equation.DSMT4 Rpta.5 En la grafica mostrada , calcular la medida el ngulo x
a) 40
b) 60
c) 80d) 50
e) 70
Resolucin:
Del grafico se tiene que:
EMBED Equation.DSMT4 Rpta.
TRINGULOS
Es la figura formada por tres segmentos de recta que se unen pos sus extremos 2 a 2.
Elementos:
Vrtices: A; B y C
Lados: AB; BC y AC
ngulos interiores:
ngulos exteriores:
Teoremas Fundamentales
1. En todo triangulo la suma de las medidas de sus ngulos interiores es 180
2. En todo tringulo la medida de un ngulo exterior es igual a la suma de las medidas de dos ngulos del tringulo no adyacentes a l.
3. En todo tringulo la suma de las medidas de sus ngulos exteriores es 360.
4. En todo tringulo la longitud de uno de sus lados est comprendido entre la suma y la sustraccin de las longitudes de los otros dos lados.
Si:
5. En todo tringulo se cumple que a mayor lado se le opone mayor ngulo y viceversa.
Si:
Clasificacin de Tringulos:
I. Por sus lados
II. Por sus ngulos
Lneas Notables en el Tringulo
Bisectriz:
Es el segmento que biseca al ngulo de referencia, se tienen bisectrices interiores y exteriores
Ceviana:
Es el segmento determinado por un vrtice y un punto cualquiera del lado opuesto o de su prolongacin.
Mediatriz:
Es la lnea recta perpendicular en el punto medio de un lado cualquiera.
Mediana:
Es el segmento determinado por un vrtice y el punto medio del lado opuesto.
Altura:
Es el segmento determinado por la partida de un vrtice y la llegada en forma perpendicular al lado opuesto o su prolongacin.
* Propiedades en el triangulo issceles.
La suma de las distancias de un punto de la base de un tringulo issceles a sus lados congruentes es igual a cualquiera de las alturas congruentes.
Consecuencia:
* Propiedades en el triangulo equiltero.
La suma de las distancias de un punto interior a un tringulo equiltero hacia sus lados es igual a cualquiera de las alturas congruentes.
ngulos Formados Por Las Lneas Notables
1. ngulo formado por dos bisectrices interiores. Su medida es igual a 90 ms la mitad de la medida del tercer ngulo interior.
2. ngulo formado por dos bisectrices exteriores. Su medida es igual a 90 menos la mitad de la medida del tercer ngulo interior.
3. ngulo formado por una bisectriz interior y una exterior, su medida es igual a la mitad del tercer ngulo interior.
Propiedades Adicionales
1.
2.
3.
4.
5.
6.
Tringulos Rectngulos Notables:
PROBLEMAS RESUELTOS
1 En un triangulo issceles ABC de base AC, sobre los lados AC y BC se ubican los puntos F y D tal que BF=BD, calcular la medida del ngulo CFD, sabiendo que el ngulo ABF=40.
Resolucin:
*
*
EMBED Equation.DSMT4 Rpta.
2 Se tiene un triangulo acutngulo ABC, donde I es el incentro y O el ortocentro y adems la medida del ngulo , calcular la medida del ngulo OBA.
a) 10
b) 22
c) 40
d) 20
e) 33
Resolucin:
En el rectngulo BHA
EMBED Equation.DSMT4 Rpta.
Congruencia de Tringulos
Primer Caso: ALA
(AnguloLadoAngulo)
Dos tringulos son congruentes si tienen congruentes un lado y los ngulos adyacentes a l.
Segundo Caso: LAL
(LadoAnguloLado)
Dos tringulos son congruentes, si tienen congruentes dos lados y el ngulo comprendido entre ellos.
Tercer Caso: LLL
(LadoLadoLado)
Cuarto Caso: LLAm
(LadoLadoAngulo mayor)
Dos tringulos son congruentes si tienen dos lados congruentes y un ngulo congruente opuesto al lado mayor.
Teorema de la base media
En todo tringulo el segmento que une los puntos medios de dos lados, es paralelo al tercer lado y su longitud igual a su mitad.
Teorema de la Bisectriz
Teorema de la Mediatriz
Los cuadrilteros, es todo polgono de cuatro lados.
CLASIFICACION:
1. Cuadriltero Convexo
Sus ngulos interiores son ngulos convexos
2. Cuadriltero Cncavo
Posee un ngulo interior cncavo
CUADRILTEROS CONVEXOS
1. PARALELOGRAMOS:
Son cuadrilteros que poseen lados paralelos dos a dos y adems congruentes entre s, entre ellos encontramos:
Cuadrado
Rectngulo
Rombo
Romboide
I CUADRADO:
Sus cuatro ngulos rectos y lados congruentes.
II RECTANGULO: Sus cuatro ngulos rectos y sus lados opuestos congruentes dos a dos.
y
III ROMBO: Sus cuatro lados congruentes y sus ngulos opuestos congruentes dos a dos.
IV ROMBOIDE: Sus lados y sus ngulos opuestos son congruentes dos a dos entre s:
PROPIEDADES DE LOS PARALELOGRAMOS
En todo paralelogramo los ngulos opuestos son congruentes.
Los ngulos adyacentes a un lado de todo paralelogramo sin suplementarios.
Las diagonales de los paralelogramos se bisecan mutuamente.
Son cuadrilteros que poseen dos lados opuestos paralelos se les denomina bases del trapecio y dos lados no paralelos.
I Trapecio Escaleno: Es el trapecio en el cual sus lados no paralelos son de diferente longitud.
II Trapecio Issceles: Sus lados no paralelos son congruentes
III Trapecio Rectngulo:
PROPIEDADES EN LOS TRAPECIOS
Los ngulos adyacentes a los lados no paralelos son suplementarios
La longitud de la mediana (MN) es igual a la semisuma de las longitudes de sus bases:
La longitud del segmento que une los puntos medios de las diagonales (PQ), es igual a la semidiferencia de longitudes de sus bases.
TRAPEZOIDES
Son cuadrilteros convexos que no poseen lados paralelos.
Se tiene dos clases de trapezoides
Trapezoide Asimtrico:
Trapezoide Simtrico:
TEOREMAS COMPLEMENTARIOS:
Si G es baricentro del triangulo
En todo Paralelogramo se cumple que:
PROBLEMAS RESUELTOS:
Ejemplo 1
En un tringulo ABC las medianas AM y BN se interceptan en el punto G, por N se traza una paralela AM que interfecta en P a la prolongacin de BA: si AB=12m y PN=AP, calcular la longitud de MG.
a) 3m
b) 5m
c) 2m
d) 4m
e) 7m
Resolucin:
Se traza: MN (base media)
De donde:
Se observa que AMNP es un rombo
En el triangulo ABC G es su baricentro, entonces diremos que:
;
EMBED Equation.DSMT4 Rpta.
Ejemplo 2
En un romboide ABCD se traza la bisectriz AE (E en BC). Si CD=6m. Calcular la longitud del segmento que une los puntos medios de AC y ED
a) 2m
b) 3m
c) 10m
d) 5m
e) 4m
Resolucin:
Se observa que
En el trapecio AECD
(Propiedad)
EMBED Equation.DSMT4 Rpta.
Ejemplo 3
En un cuadrado ABCD, sobre el lado CD se ubica un punto E de modo que AE corte a BD en F, si , Calcular:
a) 10
b) 30
c) 20
d) 40
e) 50
Resolucin:
Se observa que existe congruencia:
Entonces:
EMBED Equation.DSMT4 Rpta.
Es el conjunto de todos los puntos aferentes que constituyen una lnea curva plana y cerrada, cuyos puntos estn a la misma distancia R de un punto interior O denominado centro de la misma.
Elementos de la Circunferencia
Puntos:
O centro de la circunferencia
T punto de tangencia
punto aferente de la
circunferencia
punto interior a la circunferencia
punto exterior a la circunferencia
Rectas:
recta tangente
recta secante
recta exterior
Segmentos:
cuerda
TEOREMAS DE LA CIRCUNFERENCIA
Teorema de Poncelet
Posiciones relativas entre dos Circunferencias
Adicionales:
1. Si las circunferencias son congruentes
Concuencia:
Obseracin:
12. Si: R es punto de tangencia
Si: T es punto de tangencia
Si: T es punto de tangencia
Si: P es punto de tangencia
PROBLEMAS RESUELTOS
Ejemplo 1:
Se tienen tres circunferencias tangentes exteriores dos a dos de radios: 1m, 2m y 3m. Calcular la longitud del radio de una cuarta circunferencia que pasa por los puntos de tangencia de los tres primeros.
a) 2m
b) 1m
c) 5m
d) 3m
e) 4m
Resolucin:
Los lados del triangulo ABC con referencia a la circunferencia menor son tangentes a ella, por ende:
; ;
Los cuales determinan un triangulo rectngulo recto en A
Aplicando teorema de poncelet
EMBED Equation.DSMT4 Rpta.
Ejemplo 2:
Calcular el permetro del triangulo rectngulo, si las longitudes de los radios de las circunferencias inscrita y circunscrita miden 4m y 13m
a) 10m
b) 15m
c) 60m
d) 40m
e) n.a.
Resolucin:
Por el teorema de Poncelet
El permetro ser:
EMBED Equation.DSMT4 Rpta.
Ejemplo 3:
Sobre el lado AB de un cuadrado ABCD se construye exteriormente un tringulo rectngulo AFB, si M es el punto de interseccin de AC y BD. Calcular: .
a) 30
b) 45
c) 40
d) 50
e) n.a.
Resolucin:
Se observa que el cuadriltero AFBM es cclico (inscriptible).
EMBED Equation.DSMT4 Rpta.
Es todo conjunto de segmentos consecutivos, los cuales siguen diferentes direcciones. Es decir es toda poligonal cerrada
Clasificacin Por su Forma
1. Polgono Plano: Lados coplanares
2. Polgono Alabeado: Lados no coplanares
3 Polgono Convexo: Sus ngulos interiores son convexos.
4 Polgono Cncavo: Uno o mas ngulos interiores son cncavos.
5. Polgono equiltero: Poseen sus Lados congruentes
5 Polgono Equingulo: Poseen todos sus ngulos interiores congruentes
6. Polgono Regular: Es aquel polgono que es equiltero y equingulo a la vez
6 Polgono Irregular: Son los que poseen ngulos y lados desiguales.
POR SU NMERO DE LADOS
Triangulo
: 3 lados
Cuadriltero
: 4 lados
Pentgono
: 5 lados
Hexgono
: 6 lados
Heptgono
: 7 lados
Oc6togono
: 8 lados
Enegono
: 9 lados
Decgono
: 10 lados
Undecgono
: 11 lados
Dodecgono
: 12 lados
Pentadecgono
: 15 lados
Icosgono
: 20 lados
PROPIEDADES:
Para todo polgono de n lados:
En todo polgono numricamente: los vrtices, lados, ngulos interiores y ngulos centrales son iguales.
A partir de un vrtice de un polgono convexo se puede trazar diagonales.
El numero de diagonales, se obtiene por:
La suma de las medidas de los ngulos interiores resulta ser:
La suma de las medidas de los ngulos exteriores, resulta ser:
La suma de las medidas de los ngulos centrales
La medida de un ngulo interior de un polgono regular o equingulo:
La medida de un ngulo exterior de un polgono regular o equingulo.
PROBLEMAS RESUELTOS
1 Cual es el polgono regular en el cual al aumentar en 3 su nmero de lados, la medida de su ngulo exterior disminuye en 27?
a) Cuadriltero
b) pentgono
c) triangulo
d) hexgono
e) n.a.
Resolucin:
Por ser polgono regular
Por enunciado:
De donde:
Rpta.
2 En un heptgono, tres de sus ngulos interiores miden 120 cada uno, calcular la medida de los otros cuatro ngulos, sabiendo que son congruentes.
a) 140
b) 120
c) 135
d) 150
e) n.a.
Resolucin:
Del enunciado se tiene que:
Adems se conoce que:
EMBED Equation.DSMT4 Rpta.
3 Cuntos lados tiene el polgono convexo que al duplicar el nmero de lados, la suma de sus medidas de sus ngulos interiores se cuadruplican?
a) 2 lados
b) 4 lados
c) 3 lados
d) 5 lados
e) 6 lados
Resolucin:
Del enunciado se tiene que:
EMBED Equation.DSMT4 Rpta.
4 La diferencia entre en nmero de diagonales de un polgono regular con el nmero de ngulos rectos a que equivale la suma de las medidas de sus ngulos interiores es 8. Calcular la medida de su ngulo central?
a) 30
b) 20
c) 50
d) 45
e) 40
Resolucin:
Del enunciado se tiene:
Resolviendo:
Luego la medida de un ngulo central
EMBED Equation.DSMT4 Rpta.
1 En un tringulo ABC, es bisectriz, F esta en . Si: AB=BF=FC, Hallar la medida del ngulo BCA.
a) 30
b) 45
c) 53
d) 37
e) 36
2 En un triangulo rectngulo si uno de los ngulos agudos mide 38, Cunto mide el ngulo que forman la altura y la mediana relativa a la hipotenusa?
a) 24
b) 20
c) 18
d) 26
e) N.A.
3 Cul es el permetro de un rectngulo que tiene de diagonal 15 m y su ancho es 75% de su largo?
a) 60
b) 48
c) 42
d) 62
e) 58
4 Los lados de un tringulo miden 7, 14 y 15 m. Cunto se debe disminuir a cada lado para que el tringulo que resulte sea tringulo rectngulo?
a) 1 m
b) 1,5 m
c) 2, 5 m
d) 2 m
e) 3,5 m
5 En un cuadrado ABCD de la do 1 m se trazan 2 cuadrantes con centros en A y C y con radio 1 m, que interceptan a la diagonal AC en P y Q, hallar .
a)
b)
c)
d)
e)
6 En el cuadriltero ABCD se cumple: , , , , , calcular .
a)
b)
c)
d)
e) N.A.
7 Si el rea de un cuadrado inscrito en la circunferencia es al rea del cuadrado inscrito en las semicircunferencias como:
a) 2:1
b) 3:2
c) 5:2
d) 4:3
e) 5:3
8 Hallar la distancia entre los puntos medios de dos caras consecutivas de un cubo, cuyas aristas miden .
a) 1
b) 1/2
c) 3/8
d) 0,75
e)
9 En una circunferencia de 12 m de radio se toma un sector circular de 60. Calcular el rea del circulo inscrito en el sector?
a)
b)
c)
d)
e)
10 En un cuadrado de 1m de lado se inscribe un octgono regular de modo que 4 de sus lados estn sobre los lados del cuadrado. Hallar el lado del octgono.
a)
b)
c)
d)
e)
11 Se tiene un slido de madera de 18 cm de arista, se pintan todas sus caras y luego se divide en cubitos de 3 cm de arista, Cuntos cubitos tienen slo una cara pintada?
a) 24
b) 36
c) 96
d) 6
e) 8
12 Cul es la diferencia de las reas de dos crculos que son tangentes interiormente si la distancia entre sus centros es 7 cm y la suma de sus circunferencias es 70 cm?
a) 245
b) 280
c) 125
d) 190
e) 260
13 Hallar el rea de un rectngulo de permetro 2p inscrito en una circunferencia de radio R.
a)
b)
c)
d)
e) N.A.
14 Hallar el rea total de un cono si el ngulo formado por la generatriz y el radio de la base mide 60 y el radio es 3cm.
a)
b)
c)
d)
e)
15 Un cilindro est lleno de agua hasta la mitad. Se suelta un pedazo metlico y el nivel de agua sube 3,5 si el volumen del cilindro es 8. Hallar el volumen del pedazo.
a) 176
b) 180
c) 192
d) 200
e) 24016 Si el radio de un circulo es incrementado en una unidad, la razn de la nueva circunferencia al nuevo dimetro es:
a)
b)
c)
d)
e)
17 Los radios de 2 circunferencias ortogonales miden 8 y 15. Determinar el rea del crculo inscrita en el tringulo que se forma al unir los centros de las circunferencias originales con uno de sus puntos de corte.
a)
b)
c)
d)
e)
18 Del punto P se trazan las secantes PFA y PHB de manera que AB es el dimetro del crculo, siendo O el centro del crculo tal que: , hallar la medida del ngulo FOH.
a) 60
b) 70
c) 80
d) 53
e) 75
19 Dado un tringulo ABC de ngulos A y C iguales a 30 y 15 respectivamente, si . Calcular:
a) 6
b) 4
c) 8
d) 2,5
e) 5,2
20 En un tringulo rectngulo: , se trazan BE y BD de modo que: , adems y . Hallar la medida del ngulo EBD.
a) 36
b) 30
c) 53
d) 60
e) 37
21 Hallar el rea mxima que puede tener un rectngulo inscrito en un tringulo ABC, AC=6 y altura , uno de los lados del rectngulo est sobre .
a)
b)
c)
d)
e)
22 Calcular el radio de la circunferencia inscrita en un rombo si sus diagonales miden: 16m y 12m.
a) 4,8
b) 5,2
c) 5,1
d) 3,2
e) 3,8
23 Hallar: , si:
a)
b)
c)
d)
e)
24 Hallar x, si:
a) 30
b) 40
c) 60
d) 37
e) 4525 En la figura ABCD es un rectngulo, . Hallar el permetro.
a) 50
b) 52
c) 54
d) 49
e) 56
26 Hallar: AM en el grafico mostrado:
a)
b)
c)
d)
e) N.A.
27 Hallar la longitud de la faja que rodea las circunferencias iguales de radio 0,5 cm.
a)
b)
c)
d)
e)
28 Un cilindro contiene agua hasta sus 3/4 partes. Qu ngulo puede inclinarse de tal manera que no se derrame?
a) 37
b) 53
c) 30
d) 45
e) 60
29 Hallar el ancho del rectngulo mostrado.
a)
b)
c)
d)
e) 2
30 Hallar: x
a) 30
b) 45
c) 37
d) 65
e) 60
31 En el triangulo. Cul de las siguientes relaciones entre los ngulos es verdadera?
a) x+z=a+b
b) y+z=a+b
c) m+x+w+n
d) x+z+n=w+c+m
e) x+y+n+=a+b+m32 Si ABCD es un cuadrado de lado 2m y el tringulo CED es equiltero. Hallar el rea del tringulo DAC?
a)
b)
c) 2,25
d)
e) 2,125
33 Del grafico mostrado, determinar el ngulo x
a) 10
b) 20
c) 15
d) 18
e) 5
34 Determinar el valor del ngulo x, Si: AB=BC
a) 12
b) 15
c) 17
d) 18
e) 2035 En el grafico mostrado determinar el valor del ngulo x, si: CP=2UN
a) 36
b) 45
c) 60
d) 30
e) 18
36 Determinar el valor del ngulo .
a) 30
b) 60
c) 45
d) 37
e) 53
37 Si se tiene que: DE=EF=BE=FC y AB=AF, determinar el ngulo x.
a) 34b) 54
c) 37
d) 30
e) 60
38 Dado el triangulo ABC, se tiene que AB=8 y , determinar el mximo valor entero que puede asumir BC.a) 31
b) 21
c) 19
d) 32
e) 22
39 Dado el triangulo rectngulo ABC, recto en B, se ubican los puntos D y E exteriores y relativo a la hipotenusa; de tal forma que los tringulos AEC y BCD resulten equilteros, hallar la distancia de E a si AB=10.
a) 10 b) 5
c) 8d) 6
e) 4
40 Hallar la medida del ngulo ACB de un tringulo ABC, si se conoce que: , y
a) 30 b) 45c) 60 d) 37e) 15
41 En el siguiente grafico determinar 3x si AC=AB+BD.
a) 55
b) 30
c) 50
d) 36e) 80
42 En un triangulo ABC se traza la mediana BM, de tal manera que . Hallar: si se conoce que .
a) 20
b) 30
c) 15
d) 45
e) 60
43 Del grafico, Calcular
a) 60
b) 40
c) 50
d) 80
e) 54
44 En la figura mostrada. Calcular
a) 40
b) 50
c) 70
d) 56
e) 30
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EMBED Equation.DSMT4
EMBED Equation.DSMT4
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EMBED Equation.DSMT4
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EMBED Equation.DSMT4
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EMBED Equation.DSMT4
EMBED Equation.DSMT4
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EMBED Equation.DSMT4
EMBED Equation.DSMT4
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