+1253 BY and CZ are angle bisectors, so by the angle bisector theorem, AZ/AC = BZ/BC and CY/BC = AY/AB. Plugging in some values from the givens gets us \(\frac{AZ}{AC} = \frac{8-AZ}{BC}\) and \(\frac{4}{BC} = \frac{6}{8}\), so BC = 16/3. Also note that AC = AY+CY = 6+4 = 10, so we have AZ/10 = 8-AZ/(16/3). Solving for AZ gets us 120/23, and since AB is 8, AB-AZ = BZ so BZ = 64/23.
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