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 #8 #, 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 #4  #.  If side AC has a length of #8  #, what is the length of side BC?

Theodore Abad · Accepted Answer

Length of side BC = 6.6667

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 / 8 = 4 / (DC)# #DC = (4*8) / 12 = 2.6667# #BC = BD+DC= 4+2.6667 =6.6667#

Theodore Abad · Answer

undefined Using the angle bisector theorem, we can find the length of side BC. Let D be the point where the angle bisector of angle A intersects side BC. According to the angle bisector theorem:

BD/DC = AB/AC

Given that AB = 12 and AC = 8, we can substitute these values into the equation:

BD/DC = 12/8

Solving for BD, we get:

BD = (12/8) * DC

Since BD + DC = BC, we can substitute BD in terms of DC into this equation:

(12/8) * DC + DC = BC

Combine like terms:

(12/8 + 1) * DC = BC

Simplify:

(12/8 + 8/8) * DC = BC

(20/8) * DC = BC

5/2 * DC = BC

Given that the distance between D and B is 4, we can equate this to DC:

DC = 4

Now, we can find BC:

BC = (5/2) * 4

BC = 10

Therefore, the length of side BC is 10.