A triangle has corners at points A, B, and C. Side AB has a length of #38 #. The distance between the intersection of point A's angle bisector with side BC and point B is #8 #. If side AC has a length of #46 #, what is the length of side BC?

Let the point where the angle bisector intersects with side BC be D

Using the angle bisector theorem, the ratio of the length of side AC to the length of side BC is equal to the ratio of the length of side AB to the length of the segment that is cut by the angle bisector on side BC. Let the length of the segment that is cut by the angle bisector on side BC be x. According to the given information, the ratio of the length of side AC to the length of side BC is ( \frac{46}{x} ), and the ratio of the length of side AB to the length of the segment that is cut by the angle bisector on side BC is ( \frac{38}{x + 8} ). Since these ratios are equal, we can set up the equation ( \frac{46}{x} = \frac{38}{x + 8} ) and solve for x.

( \frac{46}{x} = \frac{38}{x + 8} )
( 46(x + 8) = 38x )
( 46x + 368 = 38x )
( 46x - 38x = -368 )
( 8x = -368 )
( x = -46 )

Since the length of a segment cannot be negative, there seems to be a mistake in the calculations. Let's correct it.

( 46(x + 8) = 38x )
( 46x + 368 = 38x )
( 46x - 38x = -368 )
( 8x = -368 )
( x = -46 )

( x = -46 ) does not make sense since the length of a segment cannot be negative. The mistake is likely in the setup of the equation. Let's recheck it.

The correct equation should be:

( \frac{46}{x} = \frac{38}{x - 8} )

Let's solve this equation:

( 46(x - 8) = 38x )
( 46x - 368 = 38x )
( 46x - 38x = 368 )
( 8x = 368 )
( x = \frac{368}{8} )
( x = 46 )

So, the length of the segment that is cut by the angle bisector on side BC is 46. Therefore, the length of side BC is ( x + 8 = 46 + 8 = 54 ).