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In isosceles right triangle ABC, shown here, AC = BC. Point X is on side BC such that CX  = 6 and XB = 12, and Y is on side AB such that XY is perpendicular to AB. What is the ratio of BY to YA?

 

 Jun 10, 2022
 #1
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Note that triangles  BYX and   BCA  are similar by AA congruency

 

And AC = BC  so    XY = BY 

 

And since BX = 12......then   BX =  sqrt ( XY^2 + BY^2)   =  sqrt (2*BY^2)  =   BY * sqrt 2

 

So  BY =  12/sqrt 2 =  6sqrt 2

 

Likewise....since  AC = BC  then

 

BA = sqrt ( AC^2 + BC^2)  =    sqrt [ 2 * AC^2]  =  AC * sqrt 2 =   18sqrt 2

 

So  YA =   BA  -BY =  18sqrt 2 - 6sqrt 2   = 12 sqrt 2

 

So  

 

BY   /  YA =   [ 6sqrt 2 ] /   [12 sqrt 2 ] =   6/12  =  1/2  

 

cool cool cool

 Jun 10, 2022

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