MAB and MCD are two secants to the same circle. Find MD if MA = 18, AB = 12, MC: DC = 5: 7.

Let the length of the MC of the secant MD segment be 5 * X cm, then the length of the CD segment = 7 * X cm.

Then МD = 5 * X + 7 * X = 12 * X.

MВ = MA + AB = 18 + 12 = 30 cm.

By the property of the secant lines of a circle drawn from one point, the product of the length of one secant by its outer segment is equal to the product of the other secant by its outer segment.

MВ * MA = MD * MС.

30 * 18 = 12 * X * 5 * X.

540 = 60 * X2.

X2 = 540/60 = 9.

X = 3.

Then MS = 5 * 3 = 15 cm, CD = 7 * 3 = 21 cm.

МD = 15 + 21 = 36 cm.

Answer: The length of the MD secant is 36 cm.