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 #8 #. If side AC has a length of #19 #, what is the length of side BC?

Question

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 #8  #.  If side AC has a length of #19  #, what is the length of side BC?

Theodore Abad · Accepted Answer

Length of BC = 20.6667

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #12 / 19 = 8 / DC# #DC = (8*19)/ (12) = 12.6667# #BC = BD+DC= 8+12.6667 =20.6667#

Violet Aldrich · Answer

undefined Using the angle bisector theorem, we have:

$$ \frac{AC}{AB} = \frac{BC}{BD} $$

Given that $ AB = 12 $, $ AC = 19 $, and $ BD = 8 $, we can substitute these values into the equation to find $ BC $.

$$ \frac{19}{12} = \frac{BC}{8} $$

Solving for $ BC $:

$$ BC = \frac{19}{12} \times 8 $$

$$ BC = \frac{19 \times 8}{12} $$

$$ BC = \frac{152}{12} $$

$$ BC = 12.67 $$

So, the length of side BC is approximately 12.67 units.