A triangle has corners at points A, B, and C. Side AB has a length of #18 #. The distance between the intersection of point A's angle bisector with side BC and point B is #3 #. If side AC has a length of #20 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #18  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #3  #.  If side AC has a length of #20 #, what is the length of side BC?

Mason Adams · Accepted Answer

#6 1/3" units"# #"let D be the point on BC where the angle bisector"# #"intersects BC"# #"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)|)))larrcolor(blue)"to find DC"# #rArr18/20=3/(DC)# #rArrDC=(3xx20)/18=20/6=3 1/3larrcolor(blue)"cross-multiplying"# #rArrBC=BD+DC=3+3 1/3=6 1/3" units"#

Aurora Ayala · Answer

Length of #BC = color(red)(19/3)#

Given : side AB = c = 18, BD = 3 & AC = 20.

To find BC.

As per angular bisector theorem,

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

#:. DC = (BD * AC) / (AB) = (3 * 20) / 18 = 10/3#

But #BC = BD + DC = 3 + (10/3) = 19/3#

Avery Allen · Answer

undefined Using the Angle Bisector Theorem, we can find the length of side BC.

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

According to the Angle Bisector Theorem, $BD/DC = AB/AC$.

Given $AB = 18$ and $AC = 20$, we can substitute these values:

$BD/DC = 18/20$

$BD/DC = 9/10$

Given that $BD + DC = BC$, and we know $BD$ (3) and the ratio $BD/DC$, we can solve for $BC$.

$3 + 3x = BC$, where $x$ represents the common multiplier for $BD$ and $DC$, found from the ratio $9/10$.

Solving for $x$:

$3x = BC - 3$

$x = (BC - 3)/3$

From the ratio $BD/DC = 9/10$, we have $BD = 3$, $DC = 3x$, and $BD/DC = 9/10$.

Substituting the values:

$3 / (3x) = 9/10$

Solving for $x$:

$x = 10/3$

Substitute $x$ back into the equation for $BC$:

$BC = 3 + 3(10/3)$

$BC = 3 + 10$

$BC = 13$

So, the length of side BC is 13.