Triangle Legs D34139

1. **State the problem:** We need to find the lengths of the legs $XZ$ and $YZ$ of right triangle $XYZ$ with coordinates $X(-2,4)$, $Z(3,4)$, and $Y(3,-2)$. 2. **Recall the formula for distance between two points:** For points $(x_1,y_1)$ and $(x_2,y_2)$, the distance is $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$ 3. **Find length of $XZ$:** Since $X$ and $Z$ have the same $y$-coordinate ($4$), $XZ$ is horizontal. The length is the difference in $x$-coordinates: $$XZ = |3 - (-2)| = |3 + 2| = 5.$$ 4. **Find length of $YZ$:** Since $Y$ and $Z$ have the same $x$-coordinate ($3$), $YZ$ is vertical. The length is the difference in $y$-coordinates: $$YZ = |4 - (-2)| = |4 + 2| = 6.$$ 5. **Summary:** The legs of the right triangle are - $XZ = 5$ units - $YZ = 6$ units These lengths correspond to the horizontal and vertical legs of the right triangle $XYZ$.