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# In the figure, point O is the center of the circle, the measure of angle RTB is 28

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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 four times the measure of angle SOT. What is the measure of minor arc RS, in degrees? Nov 12, 2020

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Call the point  where  BT  intersects the circle, A

And  we have  that

Measure of angle RTB  =  (1/2)  ( measure of minor arc RB - measure of minor arc SA)

But angle SOT  = measure of angle  SOA

And since SOA  is a  central angle, it has  the same measure as minor arc SA

And.....since ROB is also a central angle, it has the same measure as minor arc RB

Call the meaure of  minor arc SA =  angle SOT  = angle SOA  =  M

And call the measure of minor arc RB  = angle ROB =  4M

So  we have this equation

28  = (1/2) (4M - M)

56  = 3M

56/3  =  M

So  ,minor arc  SA = (56/3)°

And minor  arc RB   =  4 (56/3)  = (224/3)°

And  minor arc RS  =   180  - 56/3 - 224/3   = (260/3)° ≈  86.66°   Nov 12, 2020
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Important: When a tangent and a secant, two secants, or two tangents intersect outside a circle then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

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∠SOT = xº         ∠ROB = (4x)º            ∠RTB = 28º

∠RTB = 1/2 (∠ROB - ∠SOT)

28º = 1/2 (4x - x)            x = 18.6666666

Arc RS = 180 - 5x = 86.6666666º

Nov 13, 2020