To determine the length of the altitude to the hypotenuse in a right triangle where the hypotenuse is divided into segments of lengths 5 and 15, we use a specific relationship.
In a right triangle, the altitude to the hypotenuse creates two smaller right triangles that are similar to each other and to the original triangle. A known property in geometry is that the altitude from the right angle to the hypotenuse is the geometric mean of the two segments of the hypotenuse.
Given that the segments of the hypotenuse are 5 and 15, we can calculate the altitude (h) using the formula for the geometric mean:
[tex]\[ h = \sqrt{\text{segment1} \times \text{segment2}} \][/tex]
Substitute the given segment lengths into the formula:
[tex]\[ h = \sqrt{5 \times 15} \][/tex]
[tex]\[ h = \sqrt{75} \][/tex]
The simplified form of the square root of 75 is:
[tex]\[ h = \sqrt{25 \times 3} \][/tex]
[tex]\[ h = \sqrt{25} \times \sqrt{3} \][/tex]
[tex]\[ h = 5\sqrt{3} \][/tex]
So, the length of the altitude is \( 5\sqrt{3} \).
Therefore, the best answer from the choices provided is:
B. [tex]\( 5\sqrt{3} \)[/tex]
