A triangle has corners at points A, B, and C. Side AB has a length of #42 #. 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 #45 #, what is the length of side BC?

1 Answer
Jim G.
Jan 31, 2017

#BC=18 9/14" units" ≈18.643" to 3 dec. places"#

Explanation:

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

Then BC = BD + CD

We know that BD = 9 and have to find CD

Using the #color(blue)"Angle bisector theorem"#

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

Substitute known values into this equation.

#rArr9/(CD)=42/45#

#color(blue)"cross multiplying"#

#rArr42CD=9xx45#

#rArrCD=(9xx45)/42=(cancel(9)^3xx45)/cancel(42)^(14)=135/14=9 9/14#

#rArrBC=9+9 9/14=18 9/14≈18.643" to 3 dec.places"#

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