To find the scale factor for dilating an isosceles right triangle with a hypotenuse of 18 yards to a new hypotenuse of 12 yards, we can use the formula for the scale factor, which is:
\[ \text{Scale Factor} = \frac{\text{New Size}}{\text{Original Size}} \]
Substituting in the values:
\[ \text{Scale Factor} = \frac{12 \text{ yards}}{18 \text{ yards}} = \frac{12}{18} = \frac{2}{3} \]
Thus, the scale factor you should use is \( \frac{2}{3} \).