Question

13.x=
14.x=
y=
9
3
3
4

Explanation:

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

For a right - triangle with an altitude \(x\) drawn to the hypotenuse, the formula is \(x=\sqrt{ab}\), where \(a\) and \(b\) are the segments of the hypotenuse.
For problem 13:
\(a = 9\), \(b=3\)
\(x=\sqrt{9\times3}\)

Step2: Calculate the value

\(x=\sqrt{27}=3\sqrt{3}\)

For problem 14:
First, find \(x\) using \(x = \sqrt{3\times4}=\sqrt{12}=2\sqrt{3}\)
Then, find \(y\) using the formula \(y=\sqrt{3\times(3 + 4)}\) (another part of the geometric mean theorem: if \(y\) is a leg of the right - triangle and the segments of the hypotenuse are \(m = 3\) and \(n=4\), then \(y=\sqrt{m(m + n)}\))
\(y=\sqrt{3\times7}=\sqrt{21}\)

Answer:

1. \(x = 3\sqrt{3}\)
2. \(x=2\sqrt{3}\), \(y = \sqrt{21}\)