Question

AD and MN are chords that intersect at point B.

Answer 1

When 2 chords intersect in a circle, the product of their segments are equal.

9 * (x +1) = 15 * (x -1)

9x +9 = 15x -15

24 = 6x

x = 4
SO,
Line AD = 9 + x + 1
Line AD = 14

Line MN = 15 + x -1
Line MN = 18

Answer 2

we know that

When two chords intersect each other inside a circle, the products of their segments are equal (Intersecting Chord Theorem)

so

In this problem

Step $1$

Find the value of x

$AB*BD=MB*BN\\ 9*(x+1)=(x-1)*15\\ 9x+9=15x-15\\ 15x-9x=9+15\\ 6x=24\\ x=24/6\\ x=4$

Step $2$

Find the value of MN

we know that

$MN=MB+BN\\ MN=(x-1)+15\\ MN=(4-1)+15\\ MN=3+15\\ MN=18$

therefore

the answer is

the length of line segment MN is $18 units$