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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 5 and 12.

Use the geometric mean to solve for the value of x.

(1 point)
Responses

17
17

60
60

2.65
2.65

7.75
7.75
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Answer 1

To find the length of the line \( x \) using the geometric mean theorem, we can apply the formula where the length of the segment \( x \) from the vertex to the hypotenuse is given by:

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

where \( a \) and \( b \) are the segments of the hypotenuse created by the line \( x \). In this case, the hypotenuse is divided into segments of lengths 5 and 12.

Thus,

\[ x = \sqrt{5 \cdot 12} \]

Calculating this gives:

\[ x = \sqrt{60} \]

Now, simplifying \( \sqrt{60} \):

\[ \sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15} \approx 2 \cdot 3.872 = 7.744 \]

Rounding this value gives approximately \( 7.75 \).

So, the correct answer is:

7.75