Question

1. 18 15 x

Explanation:

Step1: Use the geometric mean theorem (altitude-on-hypotenuse theorem)

In a right - triangle, if an altitude is drawn from the right - angle vertex to the hypotenuse, then the length of each segment of the hypotenuse is the geometric mean of the lengths of the adjacent side and the hypotenuse. Also, the altitude \(h\) to the hypotenuse of a right - triangle with legs \(a\) and \(b\) satisfies the formula \(h^{2}=xy\) (where \(x\) and \(y\) are the segments of the hypotenuse). Another way is to use the fact that if we consider two similar right - triangles formed by the altitude.
Let the right - triangle have legs \(a = 18\) and \(b\) (the other leg), and the two segments of the hypotenuse be \(15\) and \(x\).
We know that in a right - triangle with an altitude \(h\) to the hypotenuse, the following proportion holds: \(\frac{18}{15 + x}=\frac{15}{18}\) (using the similarity of right - triangles). Cross - multiply gives us \(18\times18 = 15\times(15 + x)\).

Step2: Expand and solve the equation

Expand the equation: \(324=225 + 15x\).
Subtract \(225\) from both sides: \(15x=324 - 225\).
\(15x = 99\).
Divide both sides by \(15\): \(x=\frac{99}{15}=\frac{33}{5}=6.6\).

Answer:

\(x = 6.6\)