To determine which proportion satisfies the geometric mean (altitude) theorem in a triangle, let's first understand the theorem.
The geometric mean theorem (altitude theorem) states that in a right triangle, the altitude to the hypotenuse creates two smaller right triangles. This theorem can be stated as follows:
The length of the altitude is the geometric mean (mean proportional) between the two segments of the hypotenuse that it creates. If we consider [tex]\(a\)[/tex] and [tex]\(b\)[/tex] to be the segments of the hypotenuse, and [tex]\(h\)[/tex] to be the altitude to the hypotenuse, then the theorem states that:
[tex]\[ h = \sqrt{a \cdot b} \][/tex]
This can be rewritten in the form of proportions:
[tex]\[ \frac{a}{h} = \frac{h}{b} \][/tex]
Now let's check the given options one by one to see which one satisfies this form:
1. [tex]\( \frac{2}{n} = \frac{3}{m} \)[/tex]
Here, the first part [tex]\( \frac{2}{n} \)[/tex] does not match the geometric mean form [tex]\( \frac{a}{h} = \frac{h}{b} \)[/tex], so this is not the correct proportion.
2. [tex]\( \frac{2}{n} = \frac{3}{n} \)[/tex]
This proportion implies that [tex]\( 2 = 3 \)[/tex], which is not possible, so this cannot satisfy the geometric mean theorem.
3. [tex]\( \frac{2}{n} = \frac{\pi}{n} \)[/tex]
This proportion would imply [tex]\( 2 = \pi \)[/tex], which again is not possible, so this also does not satisfy the geometric mean theorem.
4. [tex]\( \frac{2}{n} = \frac{n}{3} \)[/tex]
In this case, comparing it to the form [tex]\( \frac{a}{h} = \frac{h}{b} \)[/tex]:
- Take [tex]\( a = 2 \)[/tex], [tex]\( h = n \)[/tex], and [tex]\( b = 3 \)[/tex].
- Then [tex]\( \frac{a}{h} = \frac{2}{n} \)[/tex] and [tex]\( \frac{h}{b} = \frac{n}{3} \)[/tex].
This proportion does match the geometric mean form [tex]\( \frac{2}{n} = \frac{n}{3} \)[/tex], so this is the correct proportion.
Thus, the correct option that satisfies the geometric mean (altitude) theorem for this triangle is:
[tex]\[ \frac{2}{n} = \frac{n}{3} \][/tex]
