1. **State the problem:** We are given the coordinates of triangle UVW and need to find the slopes and lengths of its sides, then classify the triangle. 2. **Recall formulas:** - Slope between points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$m=\frac{y_2-y_1}{x_2-x_1}$$ - Length between points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ 3. **Given slopes:** - $m_{UV}=6$ - $m_{VW}=\frac{5}{7}$ - $m_{WU}=-2$ 4. **Given lengths:** - $|UV|=\sqrt{37}$ - $|VW|=\sqrt{37}$ - $|WU|=5\sqrt{2}$ 5. **Check triangle type:** - Since $|UV|=|VW|$, two sides are equal, so the triangle is isosceles, not scalene. - To check if right triangle, check if slopes are negative reciprocals (perpendicular): - $m_{UV}=6$ and $m_{WU}=-2$; $6 \times (-2) = -12 \neq -1$ - $m_{VW}=\frac{5}{7}$ and $m_{WU}=-2$; $\frac{5}{7} \times (-2) = -\frac{10}{7} \neq -1$ - $m_{UV}=6$ and $m_{VW}=\frac{5}{7}$; $6 \times \frac{5}{7} = \frac{30}{7} \neq -1$ No pair of sides is perpendicular, so the triangle is not right. 6. **Final classification:** Triangle UVW is isosceles and not right.