The circumference of the chord is 72, and the distance from the center of the circle to this chord is 27. Find the diameter of the circle

Denote the key points as shown. Draw a segment of AO.
Consider the triangle AOB.
This triangle is right-angled, since the distance OB is the height (shortest distance).
AB is equal to half the length of the chord (according to the third property of the chord).
Then, by the Pythagorean theorem:
AO2 = OB2 + AB2
AO ^ 2 = 27 ^ 2 + (72/2) ^ 2
AO ^ 2 = 729 + 1296 = 2025
AO = 45 is the radius of the circle, therefore, the diameter D = 2 * AO = 2 * 45 = 90
Answer: D = 90