ANSWER
z = 22.5
Step-by-step explanation
We want to use the geometric mean theorem to find the value of z.
The geometric mean theorem for similar triangles states that:

This is gotten by equating the ratio of corresponding sides of similar triangles.
This implies that:
![\begin{gathered} \text{Cross}-mu\text{ltiplying:} \\ z\cdot\text{ z = x(x - 24)} \\ \Rightarrow z^2\text{ = 37.5(37.5 - 24)} \\ z^2\text{ = 37.5 }\cdot\text{ }13.5\text{ = }506.25 \\ z\text{ = }\sqrt[]{506.25} \\ z\text{ = }22.5\text{ } \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/ktv6vgxn8ar4e9ji7xn4pm5p7goitsw1lc.png)
That is the value of z.