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    el ángulo de 80° es un ángulo central, ¿cuánto mide el ángulo a?

    Santiago

    Chicos, ¿alguien sabe la respuesta?

    obtenga el ángulo de 80° es un ángulo central, ¿cuánto mide 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.

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    Relacion entre Angulos

    Esta información debe ser copiada en libreta:

    Un ángulo inscrito que abarca el mismo arco que un ángulo central mide la mitad que este. por ejemplo:

    El ángulo de color verde o central mide 80°, y este abarca el mismo arco que el ángulo rojo, por lo tanto el ángulo rojo o inscrito mide la mitad, siendo su medida de 40°

    Igualmente se puede decir, que el angulo central de 80° es el doble de medida que el angulo inscrito de 40°, esto dependera del valor del angulo que conozcas.

    Video

    Relación entre ángulos.

    Imagen

    Otros Tipos de ángulos (copiar en libreta)

    Falso o verdadero 60 Segundos

    ¿La medida de los ángulos es correcta según lo visto en clase?

    Opciones de respuesta

    Falso Verdadero Única respuesta 120 Segundos

    ¿Cual es la medida del angulo central?

    Opciones de respuesta

    a Miden lo mismo b 70° c 17.5°

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    Ir a TOMi Pro Única respuesta 120 Segundos

    Si el angulo central (verde) Mide 90° ¿Cuanto mide el angulo inscrito?

    Opciones de respuesta

    a Lo mismo 90° b El doble 180° c La mitad 45°

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    fuente : brainly.lat

    ¿Quieres ver la respuesta o más?
    Santiago 12 day ago
    4

    Chicos, ¿alguien sabe la respuesta?

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