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 #27 #, 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 #27 #, what is the length of side BC?

Evelyn Arboleda · Accepted Answer

The length of side BC is 26

Let D be the intersection of point A's angle bisector with side BC. Then #BD=8# The bisector theorem states that: #(AC)/(AB)=(CD)/(BD)# Then #27/12=(CD)/8# that's #CD=(27*8)/12=18# Then #BC=CD+BD=8+18=26#

Savannah Aguilar · Answer

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

Let's denote the length of side BC as x.

Using the angle bisector theorem, we can set up the following proportion:

$$\frac{AC}{AB} = \frac{BC}{BC+8}$$

Substituting the given values:

$$\frac{27}{12} = \frac{x}{x+8}$$

Now, we can solve for x:

$$27(x + 8) = 12x$$

$$27x + 216 = 12x$$

$$27x - 12x = -216$$

$$15x = -216$$

$$x = \frac{-216}{15}$$

$$x = -14.4$$

However, the length of a side cannot be negative, so we discard this solution.

Therefore, there might be a mistake in the setup of the problem. Double-checking the given information is recommended.