A triangle has corners at points A, B, and C. Side AB has a length of #16 #. The distance between the intersection of point A's angle bisector with side BC and point B is #12 #. If side AC has a length of #27 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #16  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #12  #.  If side AC has a length of #27 #, what is the length of side BC?

Savannah Aguilar · Accepted Answer

#32 1/4" units"#

Let D be the point on BC where the angle bisector from A, intersects BC

#rArrBC=BD+DC#

We know BD = 12 and require to find DC

Applying the #color(blue)"Angle bisector theorem"# to the triangle.

#color(red)(bar(ul(|color(white)(2/2)color(black)((AB)/(AC)=(BD)/(DC))color(white)(2/2)|)))#

Substitute known values into the equation.

#rArr16/27=12/(DC)#

#color(blue)"cross multiply"#

#rArr16DC=12xx27#

#rArrDC=(12xx27)/16=20 1/4#

#rArrBC=12+20 1/4=32 1/4" units"#

Aurora Ayala · Answer

undefined To find the length of side BC in the triangle ABC, we can use the Angle Bisector Theorem. According to the theorem, in a triangle, if a line bisects an angle, it divides the opposite side into segments that are proportional to the lengths of the other two sides.

Let D be the intersection point of the angle bisector of angle A with side BC. Then, according to the theorem:

BD / DC = AB / AC

Given that AB = 16 and AC = 27, and BD is given as 12, we can solve for DC:

12 / DC = 16 / 27

Cross-multiplying:
12 * 27 = 16 * DC
DC = (12 * 27) / 16
DC ≈ 20.25

Since BD + DC = BC, we can find BC:
BC = BD + DC
BC = 12 + 20.25
BC ≈ 32.25

Therefore, the length of side BC is approximately 32.25.