what is the length of the shorter of the two chords shown?

For this case, we can use the Intersecting Chord Theorem.
The theorem states that QA * AS is always equal to TA * AR.
Therefore we can write it as:
9 * 4x = 12 * (x + 2)
Performing the operations:
36x = 12x + 24
24x = 24
x=1
Calculating the length of chords QS and TR:
QS = 9 + 4x = 9 + 4 = 13
TR = 12 + (x+2) = 12 + 3 = 15
Therefore the length of the shorter chord is 13.