Question

What is the measure of arc RST?51 degrees204 degrees180 degrees102 degrees

Answer 1

Given the diagram in the question, we can draw lines from points R and T to the center of the circle as shown below:

Given that the radius of a circle bisects the tangent perpendicularly, we have that:

$\angle ORm=90\degree$

Therefore, we have:

$\angle TRO=\angle TRm-\angle ORm$

Given:

$\angle TRm=102\degree$

Then:

$\angle TRO=102-90=12\degree$

Using an isosceles triangle, we have that:

$\begin{gathered} \angle TRO+\angle OTR+\angle ROT=180\degree \\ \angle TRO=\angle OTR=12\degree\text{ (Base angles of isosceles triangle)} \end{gathered}$

Therefore, we have:

$\begin{gathered} 12+12+\angle ROT=180\degree \\ \angle ROT=180-12-12 \\ \angle ROT=156\degree \end{gathered}$

This is the arc angle of RT.

Therefore, the measure of arc RST can be gotten by using the angles at a point, so that we have:

$\begin{gathered} \angle RST=360-\angle ROT \\ \angle RST=360-156 \\ \angle RST=204\degree \end{gathered}$

The correct option is the SECOND OPTION.