Question

Use the image to answer the question.

A line, x, divides a right triangle from 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 3 and 4.

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 a right triangle divided by a line drawn from the vertex opposite the hypotenuse to the hypotenuse itself, we can use the formula:

\[
x^2 = a \cdot b
\]

where \( a \) and \( b \) are the segments into which the hypotenuse is divided.

In this case:
- One segment is 3 (let's call it \( a \))
- The other segment is 4 (let's call it \( b \))

We can substitute these values into the equation:

\[
x^2 = 3 \cdot 4
\]
\[
x^2 = 12
\]

Now, we take the square root of both sides to find \( x \):

\[
x = \sqrt{12}
\]
\[
x = 2\sqrt{3}
\]

Calculating \( 2\sqrt{3} \):

Using the approximate value \( \sqrt{3} \approx 1.732 \):

\[
x \approx 2 \cdot 1.732 \approx 3.464
\]

Now, rounding to the nearest tenth:

\[
x \approx 3.5
\]

Thus, the value of \( x \) rounded to the nearest tenth is:

\[
\boxed{3.5}
\]