In the figure, point $O$ is the center of the circle, the measure of angle $RTB$ is 29 degrees, and the measure of angle $ROB$ is three times the measure of angle $SOT$. What is the measure of minor arc $RS$, in degrees?
Let C be a point on line BT. We can extend SO to draw angle SOT. So, we have
Because \(\angle RTB = RB - SC = 3x - x = 2x = 29\)
Now, we know that arc RS = 180 - 4x = 180 - 58 = 122 degrees.
Let angle ROB = 3x
Let angle SOT = x
Because there are central angles, the arcs they intercept have the same measure
And we have the theorem that
angle RTB = (1/2) ( arc intercepted by ROB - arc intercepted by SOT)
29 = (1/2) ( 3x - x) multiply through by 2
58 = 3x - x
58 = 2x
x = 29
Arc ROB = 3x = 3(29) = 87
Arc SOT = x = 29
Minor arc RS = 180 - 87 - 29 = 64°