A triangle has corners at points A, B, and C. Side AB has a length of #6 #. 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 #6  #.  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?

Sadie Allen · Accepted Answer

#28/3=9 1/3" units"# #"let D be the point on BC where the angle bisector"# #"intersects BC"# #rArrBC=BD+DC# #BD=4" require to find "DC# #"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)|)))# #rArr6/8=4/(DC)larrcolor(blue)" cross-multiply"# #rArr6DC=8xx4# #rArrDC=(8xx4)/6=32/6=16/3# #rArrBC=4+16/3=28/3=9 1/3" units"#

Aurora Ayala · Answer

undefined Using the Angle Bisector Theorem, we can find the length of side BC. According to the theorem, the ratio of the lengths of the segments formed by the angle bisector of a triangle is equal to the ratio of the lengths of the sides of the triangle.

Let D be the point where the angle bisector of angle A intersects side BC.

Using the given information:
AD/DB = AC/CB

Substituting the given values:
(4 + x) / (6 - x) = 8 / 6

Cross multiplying:
8(6 - x) = 6(4 + x)

Expanding and simplifying:
48 - 8x = 24 + 6x

Bringing like terms to one side:
48 - 24 = 8x + 6x
24 = 14x

Dividing both sides by 14:
x = 24 / 14
x = 12 / 7

So, the length of side BC is:
BC = 6 - x
BC = 6 - 12/7
BC = (42 - 12) / 7
BC = 30 / 7