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# Someone plz help me with geo

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In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 8$, $AY = 6$, and $CY = 3$, find $BC$. [asy] pair A,B,C,X,Y,Z,I; A= (0,0); B = (1,0); C = (0.8,0.7); X = intersectionpoint(B--C , A -- (bisectorpoint(B,A,C))); Y = intersectionpoint(A--C, B -- scale(6)*( (bisectorpoint(C,B,A)) - B)); I = intersectionpoint(A--X, B--Y); Z = extension(A,B,C,I); draw(A--B--C--cycle); draw(B--Y); draw(C--Z); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$I$",I,SW); label("$Y$",Y,NW); label("$Z$",Z,S); [/asy]

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 8$, $AY = 6$, and $CY = 3$, find $BZ$. [asy] pair A,B,C,X,Y,Z,I; A= (0,0); B = (1,0); C = (0.8,0.7); X = intersectionpoint(B--C , A -- (bisectorpoint(B,A,C))); Y = intersectionpoint(A--C, B -- scale(6)*( (bisectorpoint(C,B,A)) - B)); I = intersectionpoint(A--X, B--Y); Z = extension(A,B,C,I); draw(A--B--C--cycle); draw(B--Y); draw(C--Z); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$I$",I,SW); label("$Y$",Y,NW); label("$Z$",Z,S); [/asy]

Aug 10, 2022

#1
+2541
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maybe format your question properly?

Aug 10, 2022
#2
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https://artofproblemsolving.com/texer/xjsiitim

Question properly displayed here. Click "With bbcode"

Aug 10, 2022
#3
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You can update the release date yourself: log in to your GEO account, go to your Series (GSExxxx) record, and use the 'UPDATE' button which is located at the top of the page; on the 'Data Field' page, there is a 'Data Release Date' box you can use to specify the new release date.

Aug 12, 2022
#4
+2541
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By the Angle Bisector Theorem, we have $${AY \over AB} = {YC \over CB}$$. Plugging in the given info, we have $${6 \over 8} = {3 \over CB}$$, meaning $$CB = 4$$

Now, apply the Angle Bisector Theorem again: $${AZ \over AC} = {ZB \over CB}$$. Subbing in what we know, we have $${AB \over 9} = {ZB \over 4} \ \ \ (i)$$.

For simplicity, let $$AB = x$$ and $$ZB = y$$. We also know that $$x + y = 8 \ \ \ (ii)$$.

Solving this system, we find $$x = AB = {72 \over 13}$$ and $$y = ZB = \color{brown}\boxed{32 \over 13}$$

Aug 12, 2022