Questions: Sea el triángulo ABC, ¿Cuánto mide el lado AB?

Sea el triángulo ABC, ¿Cuánto mide el lado AB?
Transcript text: Sea el triángulo $A B C$, ¿Cuánto mide el lado $A B$ ?
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Solution

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Paso 1: Identificar el tipo de triángulo

El triángulo ABC tiene dos ángulos iguales de 30°. Por lo tanto, es un triángulo isósceles. En un triángulo isósceles, los lados opuestos a los ángulos iguales tienen la misma longitud.

Paso 2: Determinar la longitud de los lados iguales

Como el triángulo ABC es isósceles con ángulos iguales en A y B, los lados opuestos a estos ángulos (BC y AC) son iguales. Se nos da que BC = 2, por lo tanto, AC también mide 2.

Paso 3: Aplicar el teorema del coseno

Dado que conocemos dos lados (AC = BC = 2) y el ángulo entre ellos (C = 180° - 30° - 30° = 120°), podemos usar el teorema del coseno para encontrar la longitud del lado AB:

AB² = AC² + BC² - 2 * AC * BC * cos(C) AB² = 2² + 2² - 2 * 2 * 2 * cos(120°) AB² = 4 + 4 - 8 * (-1/2) AB² = 8 + 4 AB² = 12 AB = √12 AB = 2√3

Respuesta final

El lado AB mide 2√3.

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