Ejecicios sencillos sobre geometria y trigonometria¡¡?

Solución :

<img src="https://tex.z-dn.net/?f=%5Ctextup%7BEl%20%5C%27area%20exterior%20comprendida%20entre%20los%20puntos%20de%20tangencia%7D%5C%5C%5Ctextup%7Bse%20reduce%20a%20encontrar%20el%20%5C%27area%20del%20tri%5C%27angulo%7D%5C%5C%5Ctextup%7Bmenos%20el%20%5C%27area%20de%20lo%20sectores%20circulares%7D%5C%5C%5Ctextup%7BAdem%5C%27as%20el%20tri%5C%27angulo%20est%C3%A1%20formado%20en%20cada%20lado%20por%202%20veces%7D%5C%5C%5Ctextup%7Bel%20radio%20de%20cada%20circunferencia%2Cpor%20lo%20tanto%7D%5C%5C%5Ctextup%7Bes%20un%20equil%5C%27atero%20y%20cada%20%5C%27angulo%20mide%2060%2Centonces%3A%7D%5C%5C%5Ctextup%7Bsu%20altura%20es%3A%7D%5C%5Ch%3D%5Csqrt%7B%282r%29%5E%7B2%7D-%28r%29%5E%7B2%7D%7D%5C%5Ch%3D%5Csqrt%7B4r%5E%7B2%7D-r%5E%7B2%7D%7D%5C%5Ch%3D%5Csqrt%7B3r%5E%7B2%7D%7D%5C%5Ch%3Dr%5Csqrt%7B3%7D%5C%5C%5Ctextup%7BEl%20%5C%27area%20del%20tri%5C%27angulo%20es%3A%7D%5C%5CA%3D%5Cfrac%7B%282r%29%28r%5Csqrt%7B3%7D%7D%7B2%7D%5C%5Ch%3D%5Cfrac%7B2r%5E%7B2%7D%5Csqrt%7B3%7D%7D%7B2%7D%5C%5Ch%3Dr%5E%7B2%7D%5Csqrt%7B3%7D%5C%5C%5Ctextup%7BEl%20%5C%27area%20de%20un%20sector%20circular%20es%3A%7D%5C%5CA%3D%5Cpi.r%5E%7B2%7D%5Cfrac%7B%5Calpha%7D%7B360%7D%5C%5CA%3D%5Cpi.%28r%29%5E%7B2%7D%5Cfrac%7B60%7D%7B360%7D%5C%5CA%3D%5Cfrac%7B%5Cpi.r%5E%7B2%7D%7D%7B6%7D%5C%5C%5Ctextup%7By%20como%20son%20tres%20sectores%20circulares%20entonces%3A%7D%5C%5CA%3D3%5Cfrac%7B%5Cpi.r%5E%7B2%7D%7D%7B6%7D%5C%5CA%3D%5Cfrac%7B%5Cpi.r%5E%7B2%7D%7D%7B2%7D%5C%5C%5Ctextup%7BLuego%20restando%20del%20%5C%27area%20triangular%20el%20area%20de%20los%20sectores%20circulares%7D%5C%5C%5Ctextup%7Bse%20tiene%20el%20%5C%27area%20entre%20los%20puntos%20de%20tangencia%7D%5C%5CA%3DA_%7B1%7D-A_%7B2%7D%3Dr%5E%7B2%7D%5Csqrt%7B3%7D-%5Cfrac%7B%5Cpi.r%5E%7B2%7D%7D%7B2%7D%5C%5CA%3D2r%5E%7B2%7D%5Csqrt%7B3%7D%7D-%5Cpi.r%5E%7B2%7D%7D%7B2%7D%5C%5CA%3Dr%5E%7B2%7D%28%5Cfrac%7B2%5Csqrt%7B3%7D-%5Cpi%7D%7B2%7D%29" />

24.

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<img src="https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B65%7D%7D%7B7%7D%3D%5Cfrac%7B2%5Csqrt%7B65%7D%7D%7B10%2Bx%7D%5C%5C%2810%2Bx%29%5Csqrt%7B65%7D%3D7%282%5Csqrt%7B65%7D%29%5C%5C10%5Csqrt%7B65%7D%2Bx%5Csqrt%7B65%7D%3D14%5Csqrt%7B65%7D%5C%5Cx%3D%5Cfrac%7B14%5Csqrt%7B65%7D-10%5Csqrt%7B65%7D%7D%7B%5Csqrt%7B65%7D%7D%5C%5Cx%3D%5Cfrac%7B4%5Csqrt%7B65%7D%7D%7B%5Csqrt%7B65%7D%7D%5C%5Cx%3D4" />

25.

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<img src="https://tex.z-dn.net/?f=b-a%3D%5Csqrt%7B%28x-1%29%5E%7B2%7D%2B%28x%2B1%29%5E%7B2%7D%7D%5C%5Cb-a%3D%5Csqrt%7Bx%5E%7B2%7D-2x%2B1%2Bx%5E%7B2%7D%2B2x%2B1%7D%5C%5Cb-a%3D%5Csqrt%7B2x%5E%7B2%7D%2B2%7D" />

26.

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<img src="https://tex.z-dn.net/?f=%5Ctextup%7BComo%20en%20el%20tri%5C%27angulo%20el%20%5C%27angulo%20A%20es%20el%20doble%20de%20B%2Cy%20si%20trazamos%20una%20vertical%7D%5C%5C%5Ctextup%7Bque%20parta%20de%20A%20y%20choque%20con%20el%20segmento%20BC%7D%5C%5C%5Ctextup%7Bentonces%20se%20formar%C3%A1n%20dos%20tri%5C%27angulos%20rect%5C%27angulos%20cuyos%20%5C%27angulos%20miden%7D%5C%5C%5Calpha%2B%5Calpha%2B90%3D180%5C%5C2%5Calpha%2B90%3D180%5C%5C%5Calpha%3D%5Cfrac%7B180-90%7D%7B2%7D%3D45%5C%5C%5Ctextup%7BPor%20lo%20tanto%7D%5C%5C2%5Calpha%3D90%5C%5C%5Ctextup%7B%20y%20entonces%20se%20tiene%20que%20el%20tri%5C%27angulo%20es%20rect%5C%27angulo%20y%20en%20consecuencia%3A%7D%5C%5C%5Coverline%7BBC%7D%5E%7B2%7D%3D%5Coverline%7BAB%7D%5E%7B2%7D%2B%5Coverline%7BAC%7D%5E%7B2%7D%5C%5C%5Coverline%7BBC%7D%3D%5Csqrt%7B%5Coverline%7BAB%7D%5E%7B2%7D%2B%5Coverline%7BAC%7D%5E%7B2%7D%7D" />

Saludos.