In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Y = \frac34$.  If $YZ = 30$, then what is $XY$?
I know that in a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. So, if $\tan Y = \frac34$, that means the opposite side of $Y$, which is $XZ$, is $3k$ units long, and the adjacent side of $Y$, which is $XY$, is $4k$ units long, for some constant $k$. I can use the Pythagorean theorem to find $k$. I have $XY^2 + XZ^2 = YZ^2$, or $(4k)^2 + (3k)^2 = 30^2$. Expanding and simplifying, I get $25k^2 = 900$, or $k^2 = 36$. Taking the positive square root, I get $k = 6$. Therefore, $XY = 4k = 4 \times 6 = 24$. 