A triangle has corners at points A, B, and C. Side AB has a length of #10 #. 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 #8 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #10  #.  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 #8  #, what is the length of side BC?

Savannah Aguilar · Accepted Answer

#BC=7 1/5" units"# #"let D be the point on BC where the angle bisector"# #"intersects BC"# #"using the "color(blue)"angle bisector theorem"# #color(red)(bar(ul(|color(white)(2/2)color(black)((AB)/(AC)=(BD)/(DC))color(white)(2/2)|)))larr" find DC"# #rArr10/8=4/(DC)# #rArrDC=(4xx8)/10larrcolor(blue)"cross-multiplying"# #rArrDC=32/10=3 1/5# #rArrBC=BD+DC=4+3 1/5=7 1/5" units"#

Evelyn Arboleda · Answer

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

The Angle Bisector Theorem states that in a triangle, if an angle bisector intersects the opposite side, it divides that side into segments that are proportional to the lengths of the other two sides.

Given that side AC has a length of 8, and the distance between the intersection of point A's angle bisector with side BC and point B is 4, we can set up the following proportion:

$$ \frac{AB}{AC} = \frac{BD}{DC} $$

Where:
- $ AB = 10 $ (given)
- $ AC = 8 $ (given)
- $ BD = 4 $ (given)

Let $ x $ be the length of $ DC $. Then $ BC = BD + DC = 4 + x $.

Plug in the given values into the proportion:

$$ \frac{10}{8} = \frac{4}{x} $$

Cross multiply:

$$ 10x = 8 \times 4 $$

$$ 10x = 32 $$

$$ x = \frac{32}{10} = 3.2 $$

So, $ DC = 3.2 $. Therefore, $ BC = BD + DC = 4 + 3.2 = 7.2 $.

Therefore, the length of side BC is $ 7.2 $ units.