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?

Guest Nov 12, 2020

#1**+2 **

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°

CPhill Nov 12, 2020

#2**+4 **

*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.*

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

*∠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**

jugoslav Nov 13, 2020