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In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$.  Given $AB = 12$, $AY = 12$, and $AC = 12$, find $BZ$.

 Dec 13, 2023
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Since all sides of △ABC have equal length, it is an equilateral triangle

 

Now, △ABY and △ACZ are also equilateral triangles due to the angle bisector properties. Therefore, Y and Z are the midpoints of AB and AC, respectively. This implies BY=YZ=CZ=6.

 

To find BZ, we can utilize the Pythagorean Theorem on right triangle BYZ. We have BY=6 and YZ=6, so:

BZ2=BY2+YZ2=62+62=36+36=72

 

Taking the square root of both sides:

BZ=72​

 

Simplifying further:

BZ=62​

 

Therefore, the length of BZ is 62​​

 Dec 14, 2023

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