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avatar+1066 

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 14$, $AY = 14$, and $CY = 8$, find $BZ$.

 Nov 29, 2023
 #1
avatar+1616 
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By the Angle Bisector Theorem, [ \frac{AZ}{ZB} = \frac{AC}{BC} = \frac{CY}{BY} = \frac{8}{14-Y}. ] Since AY+Y=14, [ \frac{AZ}{ZB} = \frac{8}{14-AY} = 8. ] Let ZB=x. Then AZ=8x. Since AZ+ZB=14, x+8x=14, so x=2.

Therefore, BZ=2​.

 Nov 29, 2023
 #2
avatar+1066 
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Thanks!

 Nov 29, 2023
edited by blackpanther  Nov 29, 2023

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