Angle ASB

On a circle with a radius of 10 cm and with a center S, the points A, B, and C are given so that the central angle ASB is 60 degrees and the central angle ASC is 90 degrees. Find the length of the circular arc and the amount of AB and AC offsets.

Correct answer:

o₁ =  10.472 cm
o₂ =  15.708 cm
d₁ =  10 cm
d₂ =  14.1421 cm

Step-by-step explanation:

$$\begin{gathered} r=10\text{cm} \\ \alpha =60{}^{\circ } \\ \beta =90{}^{\circ } \\ o=2\pi \cdot r=2\cdot 3.1416\cdot 10\doteq 62.8319\text{cm} \\ {o}_1=\frac{\alpha }{360}\cdot o=\frac{60}{360}\cdot 62.8319=10.472\text{cm} \end{gathered}$$

$o_2=\frac{\beta }{360}\cdot o=\frac{90}{360}\cdot 62.8319=15.708\text{cm}$

$$\begin{gathered} \sin \alpha /2=d_1/2:r \\ {d}_1=2\cdot r\cdot \sin (\alpha /2)=2\cdot 10\cdot \sin (60/2)=10\text{cm} \end{gathered}$$

$$\begin{gathered} \sin \beta /2=d_2/2:r \\ {d}_2=2\cdot r\cdot \sin (\beta /2)=2\cdot 10\cdot \sin (90/2)=10\sqrt{2}=14.1421\text{cm} \end{gathered}$$

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