To solve for the value of \( x \) using the geometric mean in the context of your right triangle, we can use the relationship established by the geometric mean theorem.
In a right triangle, if a line segment from the right angle to the hypotenuse divides it into segments of lengths \( a \) and \( b \), the length of the line segment (in this case \( x \)) is given by the geometric mean of the two segments:
\[ x = \sqrt{a \cdot b} \]
From your description, the hypotenuse is divided into two segments of lengths \( 4 \) and \( 13 \).
We need to calculate \( x \):
\[ x = \sqrt{4 \cdot 13} \]
First, multiply the two segments:
\[ 4 \cdot 13 = 52 \]
Now calculate the square root:
\[ x = \sqrt{52} \]
To round it to the nearest tenth, we first simplify \( \sqrt{52} \):
\[ \sqrt{52} = \sqrt{4 \cdot 13} = 2\sqrt{13} \]
Now we compute \( \sqrt{13} \) which is approximately \( 3.605551275 \):
\[ 2\sqrt{13} \approx 2 \cdot 3.605551275 \approx 7.21110255 \]
Rounded to the nearest tenth, we get:
\[ x \approx 7.2 \]
Thus, the value of \( x \) is:
\[ \boxed{7.2} \]