O is the Center of the circle. STU is a tangent at T. Chord BC=Chord CT;ATC=105 and CTU=40.
Calculate the size of C2
Since BC=CT and STU is tangent at T, we know that angle BCT = 90 degrees. Let x be the measure of angle BOC.
Since O is the center of the circle, angles BOC and BTC are supplementary (they add up to 180 degrees). Therefore, angle BTC = 180 - x.
Using the fact that angle ATC = 105, we can find angle ATU (the exterior angle to triangle ATC) to be 75 degrees.
Since angles ATU and CTU are on a straight line, we know that angle CTU = 180 - 75 = 105 degrees.
Now, using the fact that angle BCT = 90 degrees, we can find angle TCB to be 15 degrees (since angle BCT + angle TCB = 90).
Finally, we can use the fact that chord BC is equal in length to chord CT to find angle C2:
angle C2 = (1/2) angle BOC = (1/2)(180 - angle BTC) = (1/2)(180 - (180 - x)) = (1/2)x
To find x, we can use the fact that angles in a triangle add up to 180 degrees. Since we know angles ATC, ATU, and TCB, we can find angle ABC to be 75 - 15 = 60 degrees.
Therefore, angle BOC = 2(60) = 120 degrees.
And finally, angle C2 = (1/2)120 = 60 degrees.