Draw the segment OB as shown in the figure.
The distance from the chord AB to the tangent k parallel to it is denoted as CD.
CD = OC + OD, OC is the radius of the circle, find OD.
By the condition of the problem, k || AB. CD is perpendicular to k (by the tangent property), then CD is perpendicular to AB (since CD is secant for parallel lines, and the internal cross-angles are equal), then the triangle OBD is right-angled.
DB = AB / 2 = 80/2 = 40 (according to the second property of the chord)
OB is equal to the radius of the circle.
Then by the Pythagorean theorem:
OB2 = OD2 + DB2
85 ^ 2 = OD ^ 2 + 40 ^ 2
7225 = OD ^ 2 + 1600
OD ^ 2 = 7225-1600 = 5625
OD = 75
CD = OC + OD = 85 + 75 = 160
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