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In the diagram below, we have \(\angle ABC = \angle CAB = \angle DEB=\angle BDE.\) Given that AE=21 and ED=27, find BD and CA.


 

 Oct 13, 2019
 #1
avatar+106539 
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Since  angle ABC  = angle BDE   then  in triangle ADB,  AB  = AD  =  21 + 27  =  48

 

And  angle ABC  = angle BDE

And angle BDE  = angle BDE

 

So  by AA congruency,  triangle  DEB   is similar to triangle  DBA

 

So

 

DE / DB  =  DB/ AD

27/ DB  = DB/ 48

27 * 48  =  DB^2

1296  =  DB^2

36  =  DB

 

The EB  also =  36

 

And because  angle ABC  = angle  CAB =  angle DEB  = angle  BDE, then by AA congruency, triangle BAC is similar to triangle DEB

 

So 

 

CA / BA  = EB / DE

 

CA/ 48  = 36 / 27

 

CA / 48  =  4/3

 

CA  =  48 * 4  / 3    

 

CA =  48/3 *  4

 

CA  =  16 * 4

 

CA  =  64

 

 

cool cool cool

 Oct 13, 2019

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