Angles in the circumference

[fismaqui]

Let AB, BC be two adjacent sides of a regular 9-sided polygon inscribed in a circle of center O, as shown in the figure below.



Let M be the midpoint of AB and N the midpoint of radius OT perpendicular to BC. Determine the measurement, in degrees, of the angle OMN = alpha.

a) 20 °
b) 24 °
c) 28 °
d) 30 °
e) 32 °

 

[Deleted member 4993]

Let AB, BC be two adjacent sides of a regular 9-sided polygon inscribed in a circle of center O, as shown in the figure below.

View attachment 21925

Let M be the midpoint of AB and N the midpoint of radius OT perpendicular to BC. Determine the measurement, in degrees, of the angle OMN = alpha.

a) 20 °
b) 24 °
c) 28 °
d) 30 °
e) 32 °

If I were to this problem, I would first calculate measure of the angle MOT.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.

 

[fismaqui]

I can't even start the solution

 

[Dr.Peterson]

I can't even start the solution

Can you at least guess what angle MOT is, as suggested? How about angle AOB?

If not, please tell us what you can do, so we have a starting point.

 

[pka]

I can't even start the solution

Hints: \(m(\angle ABC)=\dfrac{2\pi}{9}\); \(m(\overline{AM})=m(\overline{MB})=m(\overline{BT})\)~ \&~ \(m(\overline{ON})=m(\overline{NT})\)

 

[Dr.Peterson]

Hints: \(m(\angle ABC)=\dfrac{2\pi}{9}\); \(m(\overline{AM})=m(\overline{MB})=m(\overline{BT})\)~ \&~ \(m(\overline{ON})=m(\overline{NT})\)

No, BT is not quite equal to AM and MB. T is on the circle, not the chord. This is more interesting than I realized.

 

[pka]

No, BT is not quite equal to AM and MB. T is on the circle, not the chord. This is more interesting than I realized.

GOOD CATCH!. I actually had in my notes that \(\{S\}=\overline{OT}\cap\overline{BC}\) so that \(m(\overline{AM})=m(\overline{MB})=m(\overline{BS})\)
Thank you.