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In right triangle \(XYZ\) with \(\angle YXZ = 90^\circ\), we have \(XY = 24\) and \(YZ = 25\). Find \(\tan Y\).

\(\begin{array}{|rcll|} \hline 24^2+XZ^2 &=& 25^2 \\ XZ^2 &=& 25^2-24^2 \\ &=& (24+1)^2 - 24^2 \\ &=& 24^2 + 48 + 1 - 24^2 \\ &=& 49 \\ &=& 7^2 \\ \mathbf{XZ} &\mathbf{=} & \mathbf{7} \\\\ \tan(Y) &=& \dfrac{XZ}{24} \\ \mathbf{\tan(Y)} &\mathbf{=} & \mathbf{\dfrac{7}{24}} \\ \hline \end{array}\)

\(\tan Y = \dfrac{\text{XZ}}{\text{XY}} = \dfrac{\sqrt{25^2 - 24^2}}{24} = \dfrac{7}{24}\)