SOLUTION: In an acute triangle ABC, the altitudes AD and CE are drawn. Find the length of the line segment DE if AB = 15 cm, BC = 18 cm, and AD = 10 cm.

Question 1080661: In an acute triangle ABC, the altitudes AD and CE are drawn. Find the length of the line segment DE if AB = 15 cm, BC = 18 cm, and AD = 10 cm.
Answer by ikleyn(52794) (Show Source): You can put this solution on YOUR website!
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In an acute triangle ABC, the altitudes AD and CE are drawn.
Find the length of the line segment DE if AB = 15 cm, BC = 18 cm, and AD = 10 cm.
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0.  Make a sketch to follow my arguments.


1.  The area of the triangle ABC is

    S =  = , 


    which gives you an equation 

    15*|CE| = 18*10.

    It implies |CE| =  = 12 cm.


2.  Then from the right-angled triangle BCE  |BE| =  =  = .


3.  From the right-angled triangle ABD  |BD| =  =  =  = 9.


4.  From the right-angled triangle ABD  cos(B) =  =  = .


5.  Now, to answer the problem's question, apply the cosine law:


     = 
 =  =  =  = .


    Hense,  |DE| =  = 10.78 (approximately).

Solved.

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