To find the length of the short leg of the original right triangle using the geometric mean, we can apply the relationship between the segments created by the altitude drawn from the right angle to the hypotenuse.
In a right triangle, if we draw an altitude (let's call it \( h \)) from the right angle to the hypotenuse, the segments created on the hypotenuse (let's call them \( a \) and \( b \)) can be used to find the lengths of the legs of the triangle (let's call them \( x \) for the short leg and \( y \) for the long leg) using the formula for the geometric mean:
\[ x = \sqrt{a \cdot h} \quad \text{and} \quad y = \sqrt{b \cdot h} \]
In this case, we know:
- The total length of the hypotenuse, which consists of \( a = 2 \) units and \( b = 30 \) units. Therefore, the total length is \( 2 + 30 = 32 \) units.
- The segments resulting from the drawing of the altitude are \( a = 2 \) and \( b = 30 \).
The relationships also state that the lengths of the legs of the triangle are proportional to the segments on the hypotenuse. The length of the legs can be expressed in a way based on the segments:
\[ x^2 = a \cdot (a + b) = 2 \cdot 32 = 64 \ y^2 = b \cdot (a + b) = 30 \cdot 32 = 960 \]
Now, taking the square root of \( x^2 \):
\[ x = \sqrt{64} = 8 \]
Thus, the length of the short leg of the original triangle is 8 units.
So the correct response is 8 units.