Prove that in every right triangle whose acute angles measure 75 and 15 degrees, the height corresponding to the hypotenuse is equal to a quarter of it.



Answer to a math question Prove that in every right triangle whose acute angles measure 75 and 15 degrees, the height corresponding to the hypotenuse is equal to a quarter of it.

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He aquí una prueba de que en todo triángulo rectángulo con ángulos agudos de 75 y 15 grados, la altura correspondiente a la hipotenusa es igual a un cuarto de la hipotenusa: **1. Identificar elementos clave:** Denotemos el triángulo rectángulo con: * A como el ángulo recto * B como el vértice del ángulo de 75 grados * C como el vértice del ángulo de 15 grados * h como la altura trazada desde B hacia el lado AC (la hipotenusa) * a como la longitud del lado AB (opuesto al ángulo de 75 grados) * c como la longitud del lado AC (la hipotenusa) **2. Relacionar ángulos y lados usando trigonometría:** Como tenemos un triángulo rectángulo y queremos encontrar la altura (h) en relación con la hipotenusa (c), podemos usar razones trigonométricas. * Conocemos un ángulo agudo (B = 75 grados) y necesitamos resolver un lado relativo a la hipotenusa. **3. Aplicar función sinusoidal:** La función seno (sin) relaciona el lado opuesto (a) con la hipotenusa (c) en un triángulo rectángulo: pecado(B) = a/c Sabemos que B = 75 grados y queremos encontrar h, pero esta ecuación nos ayuda a encontrar el lado a: a = c * sin(B) = c * sin(75°) **(Ecuación 1)** **4. Relacionar otros lados usando trigonometría:** Como el otro ángulo agudo (C) mide 15 grados, podemos encontrar el lado restante (b) usando el hecho de que la suma de los ángulos de un triángulo es 180 grados: A + B + C = 180° 90° + 75° + C = 180° C = 15° Ahora, podemos usar la función coseno (cos) para relacionar el lado b con la hipotenusa (c): porque(C) = b / c Sabemos que C = 15 grados, pero no estamos resolviendo directamente b. Esta ecuación es para referencia futura. **5. Altura relativa (h) y lado (a):** El triángulo ABC es similar a un triángulo rectángulo más pequeño formado por la altura (h), la mitad de la base (b/2) y el ángulo recto A. Estos triángulos comparten el mismo ángulo agudo B (75 grados). Como los lados correspondientes de triángulos semejantes son proporcionales: h / (b/2) = sin(B) **(Ecuación 2)** **6. Combinando ecuaciones y resolviendo h:** Queremos expresar h en términos de c. Ya encontramos a en la ecuación (1): a = c * sin(75°). Sustituya este valor de a en la ecuación (2): h / (b/2) = pecado(75°) h / [(c * cos(15°))/2] = c * sin(75°) **(sustituyendo b/c de la relación de función cos)** **7. Simplificando y aislando h:** * Simplifica el denominador: h / [c * cos(15°)/2] = 2h / c * cos(15°) * Como cos(15°) es un valor positivo (ángulo agudo), podemos multiplicar ambos lados por c * cos(15°): 2h = c * sen(75°) * cos(15°) * Sabemos que sin(75°) * cos(15°) se puede expresar como una identidad trigonométrica usando la fórmula de producto por suma: pecado(75°) * cos(15°) = (sen(90°) - pecado(15°)) * cos(15°) = cos(15°) - pecado(15°) * Sustituye esta identidad y resuelve para h: 2h = c * (cos(15°) - sen(15°)) h = c * (cos(15°) - sen(15°)) / 2 **8. Conclusión:** Dado que cos(15°) y sen(15°) son valores positivos (ángulo agudo), su diferencia es positiva. Por lo tanto, h = c * (cos(15°) - sin(15°)) / 2 representa un valor positivo que es **un cuarto de la hipotenusa (c)**. Hemos demostrado que en todo triángulo rectángulo con ángulos agudos de 75 y 15 grados, la altura correspondiente a la hipotenusa es igual a un cuarto de la hipotenusa.

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