hello
12.
from a careful observation, one would notice that line VS is the diameter of the circle and ut is the lenght of an arc. however, we have to find the angle UT
[tex]\begin{gathered} vs=180^0 \\ vs=vr+sr \\ vr=127^0 \\ 180=127+sr \\ sr=180-127 \\ sr=53^0 \end{gathered}[/tex]
note: vs is equal to 180 degrees because angle on a straight line is equal to 180 degree
[tex]\begin{gathered} ur=180^0 \\ ur=uv+vr \\ 180=uv+127 \\ uv=180-127 \\ uv=53^0 \end{gathered}[/tex]
now we can solve for angle UT easily
[tex]\begin{gathered} vs=vu+ut+ts \\ 180=53+ut+90 \\ 180=143^{}+ut \\ ut=180-143 \\ ut=37^0 \end{gathered}[/tex]
now we have the value of angle msince we know the value of angle UT, we can use this information to solve for length of the arc of UT
[tex]\begin{gathered} L_{ut}=\frac{\theta}{360}\times2\pi r \\ r=\frac{vs}{2}=\frac{12}{2}=6 \\ L_{ut}=\frac{37}{360}\times2\times3.142\times6 \\ L_{ut}=3.875m \end{gathered}[/tex]
from the calculations above, the length of the arc UT is equal to 3.87m
