1. The problem is to analyze the triangle with vertices A=(0,0), B=(4,0), and C=(0,4). 2. We start by plotting the points and connecting them to form the triangle. 3. The triangle's sides are: - AB: between (0,0) and (4,0) - BC: between (4,0) and (0,4) - CA: between (0,4) and (0,0) 4. Calculate the lengths of each side using the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 5. Length AB: $$AB = \sqrt{(4-0)^2 + (0-0)^2} = \sqrt{16 + 0} = 4$$ 6. Length BC: $$BC = \sqrt{(0-4)^2 + (4-0)^2} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$$ 7. Length CA: $$CA = \sqrt{(0-0)^2 + (4-0)^2} = \sqrt{0 + 16} = 4$$ 8. The triangle has two sides of length 4 and one side of length $4\sqrt{2}$. 9. This is a right triangle because the sides satisfy the Pythagorean theorem: $$4^2 + 4^2 = 16 + 16 = 32 = (4\sqrt{2})^2$$ 10. Therefore, the triangle is right-angled at point A. Final answer: The triangle with vertices A=(0,0), B=(4,0), and C=(0,4) is a right triangle with legs of length 4 and hypotenuse $4\sqrt{2}$.