+0  
 
0
72
1
avatar

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?

 

 Aug 2, 2021
 #1
avatar+181 
0

$BYX$ ~ $ABC$ by $AAA$ similarity. Because $ABC$ has $45-45-90$ angles it follows that $BYX$ has $45-45-90$ angles (and is isosceles).

 

The diagram below explains $45-45-90$ triangles. 

 

 

Thus, we can see that $BY = \frac{12}{\sqrt{2}} = 6\sqrt{2}$. 

 

Similarly, we can see that $BA = 18\sqrt{2}$ by triangle $ABC$. 

 

Subtracting $BC - BY = 12\sqrt{2} = AY$. 

 

Thus the ratio is $\frac{BY}{AY} = \frac{6\sqrt{2}}{12\sqrt{2}} = \boxed{\frac{1}{2}}$, which is our answer. 

 Aug 2, 2021

21 Online Users