find the values of $x$, $y$, and $z$. round to the nearest tenth, if necessary.\n$x =$\n$y =$\n$z =$

Explanation:

Step1: Identify geometric mean altitude theorem

The altitude $x$ is the geometric mean of the segments. $$x = \sqrt{8 \cdot 23}$$

Step2: Calculate the value of $x$

$$x = \sqrt{184} \approx 13.5646$$

Step3: Identify geometric mean leg theorem for $y$

The leg $y$ is the geometric mean of the hypotenuse. $$y = \sqrt{8 \cdot (8 + 23)}$$

Step4: Calculate the value of $y$

$$y = \sqrt{8 \cdot 31} = \sqrt{248} \approx 15.7480$$

Step5: Identify geometric mean leg theorem for $z$

The leg $z$ is the geometric mean of the hypotenuse. $$z = \sqrt{23 \cdot (8 + 23)}$$

Step6: Calculate the value of $z$

$$z = \sqrt{23 \cdot 31} = \sqrt{713} \approx 26.7020$$

Step7: Round values to the nearest tenth

$x \approx 13.6$, $y \approx 15.7$, $z \approx 26.7$

Answer:

$x = 13.6$ $y = 15.7$ $z = 26.7$