Use the image to answer the question.

Question

A line divides a right triangle from the vertex opposite the hypotenuse to the hypotenuse.

Use the geometric mean to find the value of x, which is the shortest leg of the large triangle. Round to the nearest tenth, if necessary.

(1 point)
Responses

4.9
4.9

8
8

2.8
2.8

15
15

Answer 1

To solve for the shortest leg \( x \) of the large triangle using the geometric mean, we first need to know the relationships in the triangle formed by the altitude from the right angle to the hypotenuse.

In a right triangle, if the altitude from the right angle splits the hypotenuse into two segments, then the length of the altitude is the geometric mean of the segments created on the hypotenuse.

Using the geometric mean theorem: \[ h^2 = p \cdot q \] where:

• \( h \) is the length of the altitude,
• \( p \) is the segment of the hypotenuse adjacent to one leg,
• \( q \) is the segment of the hypotenuse adjacent to the other leg.

Additionally, the lengths of the legs are also related to the segments of the hypotenuse: \[ x^2 = p \cdot (p + q) \quad \text{and} \quad y^2 = q \cdot (p + q) \] where \( x \) is one leg and \( y \) is the other leg.

If you provide specific values for the segments \( p \) and \( q \) or any other dimensions that are mentioned in the image, I can help compute the value of \( x \). Please share any numerical values shown in the image!