Find the Height of a 45-Degree Triangle with Given Base Segments

To find the height AD of triangle ABC, we can use the trigonometric relationship of tangent.

First, let's draw the triangle ABC with angle A at 45 degrees:

      B
     /|
    / |
   /  |
  /   |
 /    |
/     |

/ | A-------C

We are given that BD = 6 and CD = 4.

Now, let's label the height AD as 'h'. We need to find AD.

Using the tangent function, we have:

tangent(A) = opposite/adjacent

In this case, opposite = AD and adjacent = BD + CD = 6 + 4 = 10.

Therefore, we have:

tangent(45 degrees) = AD/10

As tangent(45 degrees) = 1, we can rewrite the equation as:

1 = AD/10

To solve for AD, we can multiply both sides of the equation by 10:

AD = 10

Therefore, the height AD of triangle ABC is equal to 10.