A triangle has corners at points A, B, and C. Side AB has a length of #44 #. The distance between the intersection of point A's angle bisector with side BC and point B is #24 #. If side AC has a length of #32 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #44  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #24  #.  If side AC has a length of #32 #, what is the length of side BC?

Wyatt Anderson · Accepted Answer

#(BC)=24*76/44=24*19/11~=41.45.#

Suppose that the bisector of #/_A# meets side #BC# in #D.#

So, from what is given, we conclude that in #DeltaABC#, #AB=44, AC=32, BD=24.#

Now, by the property of #/_# bisector, we have, #(AB)/(AC)=(BD)/(DC).#

Using, reverse compodundo, #(AB)/(AB+AC)=(BD)/(BD+DC)=(BD)/( BC).#

Subbing the above values, #44/(44+32)=24/(BC),# or, #(BC)=24*76/44=24*19/11~=41.45.#

Allison Allen · Answer

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

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

According to the angle bisector theorem:
$$ \frac{AC}{AB} = \frac{BC}{BD} $$

Given:
- $ AB = 44 $
- $ AC = 32 $
- $ BD = 24 $

We can plug in the values:
$$ \frac{32}{44} = \frac{x}{24} $$

Now, solve for $ x $:
$$ x = \frac{32}{44} \times 24 $$
$$ x = \frac{8}{11} \times 24 $$
$$ x = 17.4545 $$

So, the length of side BC is approximately 17.45 units.