Common chord

The common chord of the two circles, c1 and c2, is 3.8 cm long. This chord forms an angle of 47° with the radius r1 in the circle c1. An angle of 24° 30' with the radius r2 is formed in the circle c2. Calculate both radii and the distance between the two centers of the circles.

Correct answer:

r₁ =  2.7859 cm
r₂ =  2.088 cm
x =  2.9034 cm

Step-by-step explanation:

$$\begin{gathered} t=3.8\text{cm} \\ s=t/2=3.8/2=\frac{19}{10}=1.9\text{cm} \\ \alpha =47{}^{\circ } \\ \beta =24+30/60=\frac{49}{2}=24.5{}^{\circ } \\ \cos \alpha =s:r_1 \\ {r}_1=s/\cos \alpha =s/\cos 47°=1.9/\cos 47°=1.9/0.681998=2.786=2.7859\text{cm} \end{gathered}$$

$$\begin{gathered} \cos \beta =s:r_2 \\ {r}_2=s/\cos \beta =s/\cos 24.5°=1.9/\cos 24.5°=1.9/0.909961=2.088=2.088\text{cm} \end{gathered}$$

$x=r_1\cdot \sin \alpha +r_2\cdot \sin \beta =r_1\cdot \sin 47°+r_2\cdot \sin 24.5°=2.7859\cdot \sin 47°+2.088\cdot \sin 24.5°=2.7859\cdot 0.731354+2.088\cdot 0.414693=2.903=2.9034\text{cm}$

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