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.