Question

Use the image to answer the question.

A line, x, divides a right triangle and connects the vertex opposite the hypotenuse to the hypotenuse. The line x forms a second right angle, marked by a small square, where it meets the hypotenuse, and it divides the hypotenuse into 4 and 13.

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

(1 point)
x=

Answer 1

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:

\[
\boxed{7.2}
\]