A triangle has corners at points A, B, and C. Side AB has a length of #7 #. 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 #14 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #7  #.  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 #14  #, what is the length of side BC?

Mason Adams · Accepted Answer

#BC=12# #" let D be the point on BC where the angle bisector"# #"meets BC"# #"using the "color(blue)"angle bisector theorem"# #color(red)(bar(ul(|color(white)(2/2)color(black)((AC)/(AB)=(CD)/(BD))color(white)(2/2)|)))# require to find CD #rArr14/7=(CD)/4larrcolor(blue)" cross-multiply"# #rArr7CD=4xx14# #rArrCD=(4xx14)/7=8# #rArrBC=CD+BD=8+4=12#

Theodore Abad · Answer

undefined To find the length of side BC, we can use the angle bisector theorem. According to this theorem, in a triangle, the length of the side opposite a particular angle is proportional to the lengths of the other two sides.

Let's denote the length of side BC as x. According to the given information, the length of side AC is 14, and the length of side AB is 7.

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

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

Where BC' represents the distance between the intersection of point A's angle bisector with side BC and point B, which is given as 4.

Substituting the known values:

$$\frac{14}{7} = \frac{x}{4}$$

Solving for x:

$$x = \frac{14}{7} \times 4$$
$$x = 8$$

Therefore, the length of side BC is 8 units.