1. **Stating the problem:** Keegan is trying to determine if triangle LMN with vertices L(-1, 3), M(5, 5), and N(7, -1) is a right triangle by checking the slopes of sides LM and LN. 2. **What Keegan did:** He calculated the slopes: $$\text{slope of } LM = \frac{5 - 3}{5 - (-1)} = \frac{2}{6} = \frac{1}{3}$$ $$\text{slope of } LN = \frac{-1 - 3}{7 - (-1)} = \frac{-4}{8} = -\frac{1}{2}$$ 3. **Keegan's error:** He checked only the product of slopes of LM and LN to see if it equals $-1$ to determine if the triangle is right-angled. However, the right angle could be at any vertex, so he must check the slopes of all pairs of sides, not just LM and LN. 4. **Correct approach:** Calculate slopes of all three sides: - Slope of LM: $\frac{1}{3}$ (already found) - Slope of LN: $-\frac{1}{2}$ (already found) - Slope of MN: $$\text{slope of } MN = \frac{-1 - 5}{7 - 5} = \frac{-6}{2} = -3$$ 5. **Check products of slopes for perpendicularity:** - $\text{slope}_{LM} \times \text{slope}_{LN} = \frac{1}{3} \times -\frac{1}{2} = -\frac{1}{6} \neq -1$ - $\text{slope}_{LM} \times \text{slope}_{MN} = \frac{1}{3} \times -3 = -1$ (This product is $-1$) - $\text{slope}_{LN} \times \text{slope}_{MN} = -\frac{1}{2} \times -3 = \frac{3}{2} \neq -1$ 6. **Conclusion:** Since the product of slopes of LM and MN is $-1$, the angle at point M is a right angle. Therefore, triangle LMN is a right triangle. **Final answer:** Keegan's mistake was checking only one pair of sides for perpendicularity. He should check all pairs of sides' slopes to find if any two sides are perpendicular, confirming a right angle.