Geometry

To find the length of \( BD \) in right triangle \( \triangle ABC \) where \( \angle ABC = 90^\circ \), \( AB = 6 \), and \( AC = 10 \), we can use the property that the altitude to the hypotenuse of a right triangle creates two smaller right triangles that are similar to each other and to the original triangle. Let’s go through the steps:

 

1. **Calculate \( BC \) using the Pythagorean theorem**:

   \[
   BC = \sqrt{AC^2 - AB^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8.
   \]

 

2. **Use the geometric mean property**:

   Since \( AD \) is the altitude to hypotenuse \( BC \), it divides \( BC \) into segments \( BD \) and \( DC \). For a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments it creates on the hypotenuse:

   \[
   AD^2 = BD \cdot DC.
   \]

 

3. **Relate the sides using similarity**:

   From the similarity of triangles \( \triangle ABD \sim \triangle ABC \), we have:

   \[
   \frac{AB}{AC} = \frac{BD}{BC}.
   \]

   Plugging in the known values:

   \[
   \frac{6}{10} = \frac{BD}{8}.
   \]

 

4. **Solve for \( BD \)**:

   \[
   BD = \frac{6 \cdot 8}{10} = \frac{48}{10} = 4.8.
   \]

 

Therefore, the length of \( BD \) is \( \boxed{4.8} \).