SOLUTION: In triangle ABC, AB = 9, BC = 12, AC = 15, and CD is the angle bisector. Find the length of CD.

Algebra ->  Angles -> SOLUTION: In triangle ABC, AB = 9, BC = 12, AC = 15, and CD is the angle bisector. Find the length of CD.      Log On


   

Question 1087690: In triangle ABC, AB = 9, BC = 12, AC = 15, and CD is the angle bisector. Find the length of CD.
Found 2 solutions by ikleyn, addingup:
Answer by ikleyn(52878) About Me  (Show Source):
You can put this solution on YOUR website!
.
The key for solving this problem is this Theorem:


    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 On what segments the angle bisector divides the side of a triangle in this site).


For our triangle ABC it means that abs%28BD%29%2Fabs%28AD%29 = abs%28BC%29%2Fabs%28AC%29 = 12%2F15 = 4%2F5.


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%2812%5E2+%2B+9%5E2%29 = 15 ).


Finally, CD = sqrt%28%28abs%28BC%29%5E2%29+%2B+%28abs%28BD%29%5E2%29%29 = sqrt%2812%5E2+%2B+4%5E2%29 = sqrt%28144+%2B+16%29 = sqrt%28160%29 = 4%2Asqrt%2810%29.

Solved.

Also,  you have this free of charge online textbook on Geometry
    GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

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


Answer by addingup(3677) About Me  (Show Source):
You can put this solution on YOUR website!
AB = c = 9
BC = a = 12
AC = b = 15
Let CD be d, this is the length of this bisector.
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Length of angle bisector theorem:
d^2 = [ab/(a+b)^2]((a+b)^2-c^2)
where a, b, and c are the sides opposite A, B and C respectively.
Substitute with your numbers:
d^2 = [(12(15))/(12+15)^2]((12+15)^2-9^2) use your calculator and you get:
d^2 = 160
take the square root on both sides:
d = 12.65
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Did you draw it? I did, it's the first thing you should do whenever possible.