A triangle has corners at points A, B, and C. Side AB has a length of #24 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #21 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #24  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #6  #.  If side AC has a length of #21 #, what is the length of side BC?

Allison Allen · Accepted Answer

#BC = 11(1/4) or 11.25#

Angle bisector theorem states that the ratio of the length of the segment BD to the length of segment is equal to the ratio of length of side AB to the length of side AC.

#(BD)/(DC )= (AB )/ (AC)#

Given : AB = 24, AC = 21, BD = 6.

#:. DC = (BD*AC)/(AB) = (6*21)/24 = 21/4#

#BC = BD. + DC= 6+(21/4) = 45/4 = 11(1/4)#

James Atkins · Answer

undefined Using the angle bisector theorem, the ratio of the lengths of the two segments formed by the angle bisector on side AC is equal to the ratio of the lengths of the other two sides of the triangle. Therefore, $ \frac{BD}{DC} = \frac{AB}{AC} $. We can use this to find the length of BD. After finding the length of BD, we can calculate BC using the fact that BC = BD + DC.

Let's denote the length of BD as x. Then, the length of DC would be 21 - x. Using the angle bisector theorem, we have $ \frac{x}{21 - x} = \frac{24}{21} $. Solving this equation, we get $ x = 8 $. Therefore, BD has a length of 8, and DC has a length of 13. So, BC has a length of $ 8 + 13 = 21 $.