The chord AB of the circle divides the perpendicular diameter MN into segments MC and NC equal to 8 and 18. Find AB.

We denote the center of the circle by O.

Since MN is the diameter of the circle and point C belongs to the segment MN, then:

MN = MC + NC = 8 + 18 = 26.

Therefore, the radius of the circle is:

ON = OM = 1/2 * MN = 1/2 * 26 = 13.

Let’s calculate the length of the segment OC:

OC = NC – NO = 18 – 13 = 5.

Consider a right-angled triangle OCA. By the Pythagorean theorem we have:

OC ^ 2 + AC ^ 2 = OA ^ 2,

5 ^ 2 + AC ^ 2 = 13 ^ 2,

AC ^ 2 = 169 – 25 = 144,

AC = 12.

Since OA = OB, triangle ABO is isosceles.

Since OS is the height of triangle ABO, AC = CB.

Consequently,

AB = AC + CB = 2 * AC = 2 * 12 = 24.

Answer: AB = 24.