1. **State the problem:** We need to show that the points (4, 4), (3, 5), and (-1, -1) form a right-angled triangle without using the Pythagoras theorem. 2. **Approach:** Instead of using the Pythagoras theorem, we can use the concept of slopes. If two sides of the triangle are perpendicular, their slopes multiply to -1. 3. **Calculate slopes of the sides:** - Slope between (4, 4) and (3, 5): $$m_1 = \frac{5 - 4}{3 - 4} = \frac{1}{-1} = -1$$ - Slope between (3, 5) and (-1, -1): $$m_2 = \frac{-1 - 5}{-1 - 3} = \frac{-6}{-4} = \frac{3}{2}$$ - Slope between (-1, -1) and (4, 4): $$m_3 = \frac{4 - (-1)}{4 - (-1)} = \frac{5}{5} = 1$$ 4. **Check for perpendicularity:** - Check if any pair of slopes multiply to -1: $$m_1 \times m_2 = -1 \times \frac{3}{2} = -\frac{3}{2} \neq -1$$ $$m_2 \times m_3 = \frac{3}{2} \times 1 = \frac{3}{2} \neq -1$$ $$m_1 \times m_3 = -1 \times 1 = -1$$ Since $m_1 \times m_3 = -1$, the sides between points (4,4)-(3,5) and (-1,-1)-(4,4) are perpendicular. 5. **Conclusion:** The triangle formed by the points (4,4), (3,5), and (-1,-1) has a right angle at (4,4), so it is a right-angled triangle.