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    sea a un ángulo central en una circunferencia que mide 50o, determina la medida del ángulo inscrito que abarca el mismo arco de circunferencia que el ángulo a.

    Santiago

    Chicos, ¿alguien sabe la respuesta?

    obtenga sea a un ángulo central en una circunferencia que mide 50o, determina la medida del ángulo inscrito que abarca el mismo arco de circunferencia que el ángulo a. de este sitio.

    Circunferencia

    A P R E N D I E N D O

    M A T E M Á T I C A S

    A P R E N D I E N D O M A T E M Á T I C A S  

    Inicio Álgebra Trigonometría Analítica Diferencial Integral Graficador Tareas

    TRIGONOMETRÍA:

    TRIGONOMETRÍA: Los Ángulos de la circunferencia

    Podemos dibujar ángulos que se relacionen con la circunferencia. Según la posición que ocupen reciben nombres apropiados con relación a esa posición.

    Cuanto se refiere a los ÁNGULOS en la circunferencia, siempre RELACIONAMOS a éstos con los ARCOS que forman.

    1) Ángulo central: nos hemos referido a él en más de una ocasión; se trata del ángulo formado por dos radios que son sus lados y su vértice se encuentra en el centro O de la circunferencia.

    En la figura siguiente ves que el arco corresponde al ángulo central Ô que lo representamos con el acento circunflejo sobre la letra que representa el vértice del ángulo.

    El arco corresponde al ángulo central o lo que es lo mismo, la longitud del arco comprendido entre sus lados (los radios) pertenece al ángulo central y su medida es de 96º.

    Cuanto mayor es el ángulo central mayor será la longitud del arco que abarcan sus lados:

    Vas a tener en cuenta que cuando representamos con letras un ángulo, por ejemplo significa que la letra señalada con en este caso la O, nos referimos a que el vértice del ángulo se encuentra en dicha letra.

    Cuando nos refiramos a un arco entre dos puntos señalados con letras, por ejemplo: el arco entre los puntos A y B lo representamos:

    Las dos circunferencias de la última figura de igual radio, la longitud del arco vemos que están en razón directa con la medida del ángulo central: a mayor medida del ángulo central corresponde mayor longitud de arco. La longitud es decir, a 50º corresponde el arco y a 111º corresponde y puedes comprobar que a mayor ángulo central corresponde mayor longitud de arco.

    Muchas veces cuando nos referimos a las medidas de los arcos de la circunferencia hablamos de lo que miden sus longitudes en: m., dm., cm., pero también podemos referirnos a su medida en grados, minutos y segundos, incluso en radianes.

    Cuando decimos que un arco mide 75º12’13’’ quiere decir que su ángulo central tiene la misma medida.

    Las medidas de los arcos de la última figura puedes expresarlos también en grados: el arco mide 50º y arco 111º.

    Las medidas de los ángulos y arcos de una circunferencia se miden en grados, minutos y segundos.

    Problema:

    1)Una circunferencia tiene un radio de 5 m. ¿Cuánto mide un arco de esta circunferencia que corresponde a un ángulo central 60º?

    Respuesta: 5,23 m.

    Solución:

    La longitud total de la circunferencia m., corresponde a 380º

    Una longitud de………………………..X m. corresponden a 60º

    2)¿Cuál es la longitud de un arco en metros sabiendo que su ángulo central vale 65º y su radio 8 m.?Respuesta: 9,07 m.

    2) Ángulo inscrito: es el ángulo que tiene su vértice en un punto de la misma línea de la circunferencia y sus lados la cortan.

    Ves que el vértice se encuentra en el punto P de la circunferencia y los lados del ángulo inscrito cortan a la circunferencia en A y en B.

    ¿Cuál es la medida del arco correspondiente a este ángulo inscrito de 44º? Por supuesto que no se trata de la longitud del arco por que el ángulo tendría que ser central.Modo de calcular el valor de un ángulo inscrito:

    En primer lugar trazo una línea que une el punto B con el centro O, tal como lo puedes ver en la figura siguiente:

    El segmento OB y el segmento OP son iguales por tratarse del radio. Esto quiere decir que si los lados con vértice en O son iguales, los ángulos cuyos vértices están en B y en P serán iguales.

    Las medidas de estos ángulos los tienes a continuación y comprobamos que tienen 44º:

    Ahora observa bien la figura siguiente que como estudiamos con anterioridad e hicimos la demostración correspondiente sobre el valor de un ángulo exterior de un triángulo, decíamos que era igual a la suma de los otros dos ángulos interiores no adyacentes:

    El ángulo con vértice en O es igual a los valores de los ángulos cuyos vértices están en B y en P, podríamos escribir:

    Vemos que los ángulos ambos valen en nuestro ejemplo 44º.

    La igualdad podemos escribirla por ser iguales los ángulos

    Esto quiere decir que podemos escribir: y de esta igualdad despejamos :

    Comprobamos que el ángulo central en vale 88º, es decir, el doble que los ángulos inscritos y abarca el arco Esto significa que la medida del arco que abarca el ángulo o el ángulo valdrán la mitad de lo que abarca el ángulo central , es decir, .

    El valor de un ángulo inscrito es igual a la mitad del ángulo central, luego, la medida del arco correspondiente a un ángulo inscrito equivale a la mitad del arco que comprenden sus lados o a la mitad del ángulo central correspondiente.

    fuente : www.cecyt3.ipn.mx

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    fuente : brainly.lat

    Ángulos inscritos y semiinscritos de una circunferencia

    Ángulos inscritos y semiinscritos de una circunferencia

    [Calcul.] [ Anter. ] [ Indice ] [ Sigui. ]

    Introducción

    La circunferencia es un elemento auxiliar fundamental en el estudio de la Trigonometría y hay muchos resultados importantes relacionados con ella.

    En nuestro desarrollo vamos a hacer explícitos algunos de ellos que tienen un especial interés y, concretamente, vamos a hablar de la relación entre el radio y la longitud de la circunferencia, entre un ángulo central y el arco que abarca, entre ángulos centrales, inscritos y semiinscritos. Explicaremos también la idea de arco capaz que se deriva de estas relaciones. Así mismo, una conclusión inmediata a la que puedes llegar sin mucho esfuerzo es la manera de dibujar un triángulo rectángulo partiendo de una circunferencia, sin necesidad de trazar un perpendicular (tarea mucho más costosa).

    Definición de ángulo central, inscrito y semiinscrito

    [Applet] [ Zoom ] [ Latex ] [ Ayuda ]

    Definición:

    Se llama Ángulo inscrito en una circunferencia a cualquier ángulo que tenga su vértice en la circunferencia y que, cada una de las semirectas que constituyen sus lados sea secante a la circunferencia.

    (RB.1a)

    Se llama Ángulo semiinscrito en una circunferencia a cualquier ángulo que tenga su vértice en la circunferencia, una de las semirectas que determina sus lados sea tangente a la circunferencia y la otra sea secante.

    (RB.1b)

    Se llama Ángulo central en una circunferencia a cualquier ángulo que tenga su vértice en el centro de la circunferencia y que se forme con 2 semirectas con origen en él.

    (RB.1c)

    El ángulo central es proporcional al arco que abarca

    [Applet] [ Zoom ] [ Latex ] [ Ayuda ]

    En toda circunferencia, el tamaño de un arco es proporcional al tamaño del ángulo central que lo abarca.

    Como casos particulares, El arco de media circunferencia mide la mitad de la circunferencia entera ó, el arco de un cuarto de circunferencia mide la cuarta parte de la circunferencia entera.

    Dada una circunferencia L, si llamamos Lα a la longitud de un arco que es abarcado por un ángulo cental α, entonces, hay una constante t de modo que:

    Lα = t · α.

    El valor de la constante dependerá de la unidad en la que estemos midiendo los ángulos.

    (RB.2)

    Ángulos inscritos y ángulos centrales

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    En una circunferencia siempre se cumple la siguiente relación entre los ángulos inscritos y los centrales:

    "Un ángulo central β que abarca el mismo arco que un ángulo inscrito α, es el doble de α. Es decir: β = 2·α.

    (RB.3)

    La demostración, que se basa simplemente en el principio de que los ángulos de un triángulo suman dos rectos, la podemos trabajar y comprender en el Applet adjunto.

    Corolarios del resultado anterior

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    Sin más que pensar un poco, el resultado anterior nos permite concluir algunas cuestiones curiosas que puedes comprobar en el Applet adjunto:

    Cualquier ángulo inscrito que abarque media circunferencia es un ángulo recto, ya que su ángulo central correspondiente es llano (180º).

    Si trazamos un cuerda en una circunferencia, llamamos B y C a sus puntos de corte con la circunferencia y A es un punto cualquiera de la circunferencia, mientras el punto A se mueva en el mismo lado de la cuerda, el ángulo determinado por BAC, siempre valdrá lo mismo, ya que el ángulo central correspondiente no varía.

    Si trazamos un cuerda en una circunferencia que no sea un diámetro y llamamos B y C a sus puntos de corte con la circunferencia; tomando un punto cualquiera de la circunferenica a un lado de la cuerda A y otro punto al otro lado de la cuerda A', el ángulo determinado por BAC y el determinado por BA'C son suplementarios ya que sus ángulos centrales correspondientes hacen un ángulo completo (360º).

    (RB.4)

    Polígonos regulares. Ángulos

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    Definimos un polígono regular de n-lados a aquél que se obtiene llevando una distancia fija n-veces a lo largo de una circunferencia, terminando el el mimo punto que empezamos.

    Así definido un poligono regular de n-lados se puede dividir en n triángulos isósceles todos con un vértice común O el centro de la circunferencia y cada uno de ellos con un lado común con cada uno de los triángulos a su derecha e izquierda respectivamente.

    Cada ángulo central será el resultado de dividir 360º entre el número de lados del polígono.

    Cada ángulo interior es un ángulo inscrito que abarca lo mismo que el ángulo central correspondiente a n-2 lados

    Ángulos semiinscritos y ángulos centrales

    fuente : jorgefernandezherce.es

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    Santiago 12 day ago
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