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In the figure, point O is the center of the circle, the measure of angle RTB is 28 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? https://latex.artofproblemsolving.com/a/7/8/a78a6b8120f68a5e4ac574ddc1f2272768899976.png

 May 23, 2023
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To find the measure of minor arc RS, we need to determine the measure of angle SOT first.

We are given that angle RTB measures 28 degrees. Since angle RTB is an inscribed angle, its measure is half the measure of the intercepted arc RS. Therefore, the measure of intercepted arc RS is 2 * 28 = 56 degrees.

Now, let's denote the measure of angle SOT as x degrees. According to the given information, the measure of angle ROB is three times the measure of angle SOT. Therefore, angle ROB measures 3x degrees.

Since angle ROB is also an inscribed angle, its measure is half the measure of the intercepted arc RS. Thus, we have:

3x = 56 x = 56 / 3

Finally, the measure of minor arc RS is twice the measure of angle SOT, since it is intercepted by angle RTB. Thus:.

2 * 56/3 = 112/3 degrees

Therefore, the measure of minor arc RS is 112/3 degrees.

 May 23, 2023

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