A triangle has corners points A, B, and C. Side AB has a length of #15 #. 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 #18 #, what is the length of side BC?

Question

A triangle has corners points A, B, and C.  Side AB has a length of #15  #.  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 #18  #, what is the length of side BC?

James Atkins · Accepted Answer

Length of BC = 8.8

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #15 / 18 = 4 / (DC)# #DC = (4*18) / 15 = 4.8# #BC = BD+DC= 4+4.8 =8.8#

Ella Armstrong · Answer

undefined To find the length of side BC, we can use the Angle Bisector Theorem. According to the 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.

Given that the length of side AB is 15 and the length of side AC is 18, let's denote the length of side BC as $x$. According to the Angle Bisector Theorem:

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

Substituting the given values:

$$\frac{18}{15} = \frac{x}{x+4}$$

Now, cross multiply and solve for $x$:

$$18(x + 4) = 15x$$

$$18x + 72 = 15x$$

$$3x = 72$$

$$x = 24$$

So, the length of side BC is 24 units.