SOLUTION: In the diagram to the bottom, circle with center O has a radius of 8cm. Segment AT is tangent to the circle. Angle AOT=60 degrees, and AX=XY (this length is labeled m). Find the le

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Question 1148362: In the diagram to the bottom, circle with center O has a radius of 8cm. Segment AT is tangent to the circle. Angle AOT=60 degrees, and AX=XY (this length is labeled m). Find the length of m.
Diagram: https://imgur.com/a/5jS4WF7
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


With angle AOT 60 degrees and the radius OT=8, we can conclude that AO=16 and AT=8*sqrt(3).

Then in right triangle XOT we have legs XT=8*sqrt(3)-m and OT=8, and hypotenuse XO=8+m. So



Solving that equation, the m^2 terms on the two sides cancel, leaving a linear equation in m, making it possible to find an exact value for m (in radical form).
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

            There is   EXTREMELY  SIMPLE  solution. 


From the diagram,  we have an isosceles triangle AXO with equal sides lengths  | AX | = | OX |.


The base AO of this triangle is 16 units long, since it is the hypotenuse of the (30° - 60° - 90°) triangle AOT.


The angle A is 30°.


Draw the altitude  XZ  in the triangle AXO.

Then you will get right angled triangle AXZ with the long leg  AZ of the length of 8 = 16/2 units and the angle A of 30°.


Thus  cos(A) = cos(30°) =  = ,


which implies  m =  =  = 9.237 units approximately, if you want to have the numerical value.


ANSWER.  m =  = 9.237 (approximately).


Solved.


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