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 #24 #, 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 #24  #, what is the length of side BC?

Caroline Aucoin · Accepted Answer

#24" units"# #"let the point where the angle bisector intersects with"# #"side BC be D"# #"using the "color(blue)"angle bisector theorem"# #color(red)(bar(ul(|color(white)(2/2)color(black)((AB)/(AC)=(BD)/(DC))color(white)(2/2)|)))# #rArr12/24=8/(DC)larrcolor(blue)"cross-multiplying gives"# #12xxDC=8xx24# #rArrDC=(8xx24)/12=16# #rArrBC=BD+DC=8+16=24" units"#

Sophia Alfred · Answer

undefined To find the length of side BC, we can use the angle bisector theorem. According to this theorem, the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.

Let's denote the length of BC as $x$. According to the angle bisector theorem, the ratio of the length of side AC to the length of side AB is equal to the ratio of the length of segment BC to the length of segment AB.

Using the given information:
$ \frac{AC}{AB} = \frac{24}{12} = 2 $

So, we have:
$ \frac{BC}{8} = 2 $

Now, we can solve for $BC$:
$ BC = 8 \times 2 = 16 $

Therefore, the length of side BC is 16.