The radius of the circle centered at the point O is 82, the length of the chord AB is 36.

The radius of the circle centered at the point O is 82, the length of the chord AB is 36. Find the distance from the chord AB to the tangent line k parallel to it.

Let us draw from point O the radius OA to point A, the radius OB to point B and the radius to the point of tangency K. Triangle AOB is isosceles. Since AB and line k are parallel, then OK is perpendicular to AB, since it is perpendicular to the line k parallel to it. OK intersects AB at point H, OH is the height of the AOB triangle, as well as the median (since it is isosceles), therefore AH = BH = AB / 2 = 36/2 = 18.
Consider the triangle AНO: angle AНO = 90 degrees, AH = 18 – leg, OA = 82 – hypotenuse. By the Pythagorean theorem, we find the OH leg:
OH = √ (OA ^ 2 – AH ^ 2);
OH = √ (82 ^ 2 – 18 ^ 2) = √ (6724 – 324) = √6400 = 80.
The OK radius consists of two segments ОН = 80 and НК – the distance from the chord AB to the straight line k, therefore:
OK = OH + НK;
НK = OK – OH;
НK = 82 – 80 = 2.
Answer: НK = 2.
If the tangent k is located behind the center of the circle O, then the distance will be 162 (2OK – НK = 2 * 82 – 2 = 164 – 2 = 162).