The radius of the circle centered at the point O is 65; the length of the chord AB

The radius of the circle centered at the point O is 65; the length of the chord AB is 126. Find the distance from the chord AB to the parallel tangent to it k.

Let the radius OA of the circle centered at the point O be 65, the length of the chord AB is 126. If we connect the ends of the chord with the center of the circle, we get an isosceles triangle with the base AB. Let’s draw the median AK to the base of the triangle. By the property of the median of an isosceles triangle, it is also a height, which means that ΔAOK is rectangular, in it the leg AK = AB: 2; AK = 126: 2; AK = 63. Let’s find the height of AK using the Pythagorean theorem: AO² = AK² + KO²; 65² = 63² + KO²; KO = 16. The distance from the chord AB to the tangent k parallel to it, lying with it on one side of the center of the circle will be: 65 – 16 = 49. The distance from the chord AB to the tangent k parallel to it, lying with it on opposite sides from the center of the circle will be: 65 + 16 = 81.
Answer: 49 and 81.