A triangle has corners at points A, B, and C. Side AB has a length of #56 #. The distance between the intersection of point A's angle bisector with side BC and point B is #9 #. If side AC has a length of #48 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #56  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #9  #.  If side AC has a length of #48 #, what is the length of side BC?

Aaliyah Anderson · Accepted Answer

Length of side BC = 16.7143

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #56 / 48 = 9 / (DC)# #DC = (9*48) / 56 = 7.7143# #BC = BD+DC= 9+7.7143 =16.7143#

Mason Adams · Answer

undefined To find the length of side BC, we can use the Angle Bisector Theorem, which states that in a triangle, the angle bisector of an angle divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.

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

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

Given that AB = 56 and AC = 48, we can substitute these values into the equation:

$$ \frac{BD}{DC} = \frac{56}{48} = \frac{7}{6} $$

Let x be the length of BD. Then DC will be 6x/7.

Also, given that the distance between the intersection of point A's angle bisector with side BC and point B is 9, we have:

$$ BD + 9 = DC $$

Substituting the expressions for BD and DC:

$$ x + 9 = \frac{6x}{7} $$

$$ 7x + 63 = 6x $$

$$ x = -63 $$

However, the length cannot be negative, so we discard that solution.

Thus, the length of side BC is $ 6x/7 = \frac{6(-63)}{7} = -54 $.

However, since length cannot be negative, we discard this result as well.

It seems there may be a mistake in the given information or calculation.