24)
Notice that the smaller right triangle is similar to the larger right triangle.
The correspondence between the sides of the smaller triangle to the sides of the larger triangle is the following:
The side with length 4 corresponds to the side with length 12.
The side with length y corresponds to the side with length z.
The side with length 12 corresponds to the side with length x.
Use the Pythagorean Theorem applied to the smaller triangle to find the value of y:
[tex]\begin{gathered} 4^2+y^2=12^2 \\ \Rightarrow y^2=12^2-4^2 \\ \Rightarrow y^2=144-16 \\ \Rightarrow y^2=128 \\ \Rightarrow y=\sqrt[]{128} \\ \therefore y=8\cdot\sqrt[]{2} \end{gathered}[/tex]Since corresponding parts of the triangles are proportional, then:
[tex]\begin{gathered} \frac{x}{12}=\frac{12}{4} \\ \Rightarrow\frac{x}{12}=3 \\ \Rightarrow x=3\times12 \\ \therefore x=36 \end{gathered}[/tex]On the other hand:
[tex]\begin{gathered} \frac{z}{y}=\frac{12}{4} \\ \Rightarrow\frac{z}{y}=3 \\ \Rightarrow z=3y \\ \Rightarrow z=3\cdot8\cdot\sqrt[]{2} \\ \Rightarrow z=24\cdot\sqrt[]{2} \end{gathered}[/tex]Therefore, the values of x, y and z are:
[tex]\begin{gathered} x=36 \\ y=8\cdot\sqrt[]{2} \\ z=24\cdot\sqrt[]{2} \end{gathered}[/tex]