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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

2.65
2.65

17
17

60
60

7.75
7.75
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Answer 1

To solve for the value of \( x \) using the geometric mean, we can use the property of right triangles. According to the theorem regarding the altitude to the hypotenuse, the length of the altitude \( x \) can be expressed as:

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

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

Now we can calculate \( x \):

\[ x = \sqrt{5 \cdot 12} = \sqrt{60} \]

Next, we compute \( \sqrt{60} \):

\[ \sqrt{60} \approx 7.75 \]

Therefore, the value of \( x \), obtained using the geometric mean, is approximately:

7.75