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Two of the altitudes of an acute triangle divide the sides into segments of lengths 5,3,1 and x units, as shown. What is the value of x?

 

 Feb 20, 2021
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Length  of   altitude  with  an  upward  slope   =  sqrt  (8^2  - 1^2)  =  sqrt  (63)

 

Length  of  downward sloping altitude =  sqrt  [ ( x + 1)^2  - 3^2  ]  =  sqrt  [ x^2 + 2x - 8 ]

 

 

Area of  triangle  =  Area  of Triangle

 

(1/2) (3 + 5)sqrt [ x^2 + 2x - 8 ]  =  (1/2) (1 + x)  sqrt (63)

 

8 sqrt  [ x^2  + 2x  - 8 ]    =  ( 1 + x) sqrt (63)

 

Not  hard  to  solve  but tedious.......with a little  help  from WolframAlpha  

 

x = 23

 

 

cool cool cool

 Feb 20, 2021

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