A triangle has corners at points A, B, and C. Side AB has a length of #32 #. 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 #28 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #32  #.  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 #28 #, what is the length of side BC?

Avery Allen · Accepted Answer

Length of side BC = 45

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #32 / 28 = 24 / (DC)# #DC = (24*28) / 32 = 21# #BC = BD+DC= 24+21 =45#

Ella Armstrong · Answer

undefined To find the length of side BC, we can use the Angle Bisector Theorem, which states that in a triangle, the length of a side divided by the length of the corresponding segment of the opposite side (bisected by the angle bisector) is equal to the ratio of the lengths of the other two sides.

Let D be the point where the angle bisector of angle A intersects side BC. According to the Angle Bisector Theorem:

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

Given that AB = 32 and AC = 28, we can substitute these values into the equation:

$$ \frac{BD}{CD} = \frac{32}{28} $$

$$ \frac{BD}{CD} = \frac{8}{7} $$

Given that BD + CD = BC, and we know that BD = 24, we can solve for CD:

$$ 24 + CD = BC $$

$$ CD = BC - 24 $$

Substituting this expression into our ratio equation, we have:

$$ \frac{24}{BC - 24} = \frac{8}{7} $$

Cross-multiplying:

$$ 24 \times 7 = 8 \times (BC - 24) $$

$$ 168 = 8BC - 192 $$

$$ 8BC = 360 $$

$$ BC = \frac{360}{8} = 45 $$

Therefore, the length of side BC is 45 units.