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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?

 

 Feb 20, 2021
 #1
avatar+944 
+1

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.

 Feb 20, 2021
 #2
avatar+118069 
+1

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)

 

So

 

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°

 

 

cool cool cool

 Feb 20, 2021

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