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 #9 #. If side AC has a length of #11 #, 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 #9  #.  If side AC has a length of #11  #, what is the length of side BC?

Wyatt Anderson · Accepted Answer

17.25

The first step is to let D be the point on BC where the angle bisector intersects with it. Then using the #color(blue)" Angle bisector theorem "# #color(red)(|bar(ul(color(white)(a/a)color(black)( (BD)/(DC) = (AB)/(AC))color(white)(a/a)|)))# Require to find DC Substitute the appropriate values into the ratio to obtain # 9/(DC) = 12/11 # and cross-multiplying: #12xxDC = 11xx9 # To obtain DC ,divide both sides by 12 # (cancel(12) DC)/cancel(12) = (11xx9)/12 # #rArr DC = 8.25 # Now, BC = BD + DC = 9 + 8.25 = 17.25

Caroline Aucoin · Answer

undefined Using the Angle Bisector Theorem, the ratio of the lengths of the segments of side AC, split by the angle bisector from A, is proportional to the ratio of the lengths of the other two sides of the triangle. Let $x$ be the length of side BC.

According to the Angle Bisector Theorem:

$$\frac{BC}{9} = \frac{AC}{12}$$

Substituting the given values:

$$\frac{x}{9} = \frac{11}{12}$$

Cross multiply and solve for $x$:

$$12x = 9 \times 11$$
$$x = \frac{9 \times 11}{12}$$
$$x = 8.25$$

So, the length of side BC is 8.25.