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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 6 and 3.

Use the geometric mean to find x. Round your answer to the nearest tenth.

(1 point)

1 answer

To find the length of the line \( x \) that divides the right triangle, we can use the geometric mean theorem, which states that if a line is drawn from the vertex of a right triangle to the hypotenuse, it is the geometric mean of the two segments it creates on the hypotenuse.

In this case, the hypotenuse is divided into two segments of lengths 6 units and 3 units. Let's denote these segments as \( a = 6 \) and \( b = 3 \).

The length of the line \( x \) can be calculated using the formula:

\[ x = \sqrt{a \times b} \]

Substituting the values of \( a \) and \( b \):

\[ x = \sqrt{6 \times 3} = \sqrt{18} \]

Calculating \(\sqrt{18}\):

\[ \sqrt{18} \approx 4.2426 \]

Rounding to the nearest tenth:

\[ x \approx 4.2 \]

Thus, the length of line \( x \) is approximately \( 4.2 \) units.