1. **State the problem:** Determine if triangle ABC with vertices A(0,4), B(1,2), and C(4,6) is a right triangle. 2. **Recall the rule:** A triangle is right if one angle is 90 degrees. Using coordinates, this means the vectors forming that angle are perpendicular. Two vectors are perpendicular if their dot product is zero. 3. **Calculate the vectors for the sides:** - Vector AB = B - A = (1-0, 2-4) = (1, -2) - Vector BC = C - B = (4-1, 6-2) = (3, 4) - Vector CA = A - C = (0-4, 4-6) = (-4, -2) 4. **Check dot products to find perpendicular sides:** - $\vec{AB} \cdot \vec{BC} = (1)(3) + (-2)(4) = 3 - 8 = -5$ - $\vec{BC} \cdot \vec{CA} = (3)(-4) + (4)(-2) = -12 - 8 = -20$ - $\vec{CA} \cdot \vec{AB} = (-4)(1) + (-2)(-2) = -4 + 4 = 0$ 5. Since $\vec{CA} \cdot \vec{AB} = 0$, vectors CA and AB are perpendicular, so angle at point A is 90 degrees. 6. **Conclusion:** Triangle ABC is a right triangle with the right angle at vertex A.