Use the image to answer the question.

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

60
60

7.75
7.75

17

Answer 1

In a right triangle, when a line is drawn from the vertex opposite the hypotenuse to the hypotenuse, and it creates two segments on the hypotenuse (let's call them \( a \) and \( b \)), the length of the line is given by the geometric mean of the two segments into which it divides the hypotenuse.

Given:

• \( a = 5 \)
• \( b = 12 \)

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

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

Substituting in the values:

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

Calculating \( \sqrt{60} \):

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

Therefore, the value of \( x \) is approximately 7.75.

So the correct response is:

7.75