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Use the geometric mean to solve for the value of x. Round your answer to the nearest tenth.

To solve for \( x \) using the geometric mean in this right triangle configuration, we can observe that there are two right triangles sharing a common altitude (the segment labeled \( x \)), creating a geometric mean relationship. We will use the properties of right triangles and the geometric mean theorem.

The given right triangle has:
- The hypotenuse of the larger triangle as \( 13 \),
- A leg of the smaller triangle as \( 4 \),
- The altitude \( x \) that divides the hypotenuse.

### Geometric Mean Theorem

In a right triangle, if an altitude is drawn from the right angle to the hypotenuse, it divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of these two segments.

1. Let's denote:
   - The segment adjacent to the \( 4 \) as \( a \),
   - The segment adjacent to the other right angle as \( b \),
   such that \( a + b = 13 \) (the hypotenuse).

2. The geometric mean relationship states:
   \[
   x = \sqrt{a \cdot b}
   \]

### Solving for \( a \) and \( b \)

To apply this theorem, we can observe that the smaller triangle with sides \( x \) and \( 4 \) is similar to the larger triangle.

Since they are similar, we have:
\[
\frac{4}{x} = \frac{x}{b}
\]

Similarly, the triangle with sides \( x \) and the segment

Learn how to solve for x in a right triangle using the geometric mean. Given a hypotenuse of 13 and a leg of 4, find the altitude using the geometric mean theorem, suitable for students in Grades 9-10.

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