MHB Find Length of Arc EF in Triangle ABC

[Albert1]

$\triangle ABC$,with $\angle A=70^o,$point $O$ is the midpoint of segment $\overline{BC}=12$
circle $O$(wth center $O$ and radius $\overline{BO})$, meets with $\overline{AB},\overline{AC}$ at points $E$ and $F$ respectively please find the length of arc $EF$

 

[Opalg]

$\triangle ABC$,with $\angle A=70^o,$point $O$ is the midpoint of segment $\overline{BC}=12$
circle $O$(wth center $O$ and radius $\overline{BO})$, meets with $\overline{AB},\overline{AC}$ at points $E$ and $F$ respectively please find the length of arc $EF$

[sp]

View attachment 4073

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In the diagram, the angle $BFC$ is a right angle (angle in a semicircle). So $ABF$ is a right-angled triangle and $\angle ABF = 20^\circ$. Therefore $\angle EOF = 40^\circ$ (angle at centre = twice angle at circumference). If $BC$ is 12 units then the radius of the circle is 6 units, and the arc $EF$ is $2\pi\cdot 6\cdot\frac{40}{360} = \frac43\pi \approx 4.1888$ units.

[/sp]

 

[Albert1]

perfect ! very good solution !