Label 4 points on the largest right triangle, A, B, C, D. There is a theorem (right triangle altitude theorem or geometric mean theorem) that establishes similarity between the 3 triangles in the image.
[tex]\Delta ABC\sim\Delta ADB[/tex]
So corresponding sides occur proportionally to one another such that
[tex]\dfrac{AC}{AB}=\dfrac{AB}{AD}\implies\dfrac x9=\dfrac{16}x[/tex]
[tex]\implies x^2=144\implies x=12[/tex]
