Question 1087690
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The key for solving this problem is this <U>Theorem</U>:


    In a triangle, the angle bisector divides the side to which it is drawn, in two segments 
    proportional to the ratio of two other sides of a triangle.


(see the lesson <A HREF=https://www.algebra.com/algebra/homework/word/geometry/On-what-segments-the-angle-bisector-divides-the-side-of-a-triangle.lesson>On what segments the angle bisector divides the side of a triangle</A> in this site).


For our triangle ABC it means that {{{abs(BD)/abs(AD)}}} = {{{abs(BC)/abs(AC)}}} = {{{12/15}}} = {{{4/5}}}.


OK. Then, together with BD + AD = AB = 9 it means that BD = 4 and AD = 5.


Next, I am sure that you understand, without my explanations, that the triangle ABC is right-angled, since AC = {{{sqrt(12^2 + 9^2)}}} = 15 ).


Finally, CD = {{{sqrt((abs(BC)^2) + (abs(BD)^2))}}} = {{{sqrt(12^2 + 4^2)}}} = {{{sqrt(144 + 16)}}} = {{{sqrt(160)}}} = {{{4*sqrt(10)}}}.
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Solved.


Also, &nbsp;you have this free of charge online textbook on Geometry

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A> 

in this site.


The referred lesson is the part of this online textbook under the topic "<U>Properties of triangles</U>".