# A triangle has corners at points A, B, and C. Side AB has a length of #12 #. The distance between the intersection of point A's angle bisector with side BC and point B is #4 #. If side AC has a length of #9 #, what is the length of side BC?

Length of BC = 7

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

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Using the Angle Bisector Theorem, we can find the length of side BC. The theorem states that in a triangle, the length of the side that the angle bisector intersects is proportional to the lengths of the other two sides.

Let the point where the angle bisector intersects side BC be D. According to the theorem:

( \frac{BD}{DC} = \frac{AB}{AC} )

Given that AB = 12, AC = 9, and BD = 4, we can substitute these values into the equation:

( \frac{4}{DC} = \frac{12}{9} )

Now, solve for DC:

( \frac{4}{DC} = \frac{12}{9} )

( 4 \times 9 = 12 \times DC )

( 36 = 12 \times DC )

( DC = \frac{36}{12} )

( DC = 3 )

Therefore, the length of side BC is 3 + 4 = 7.

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