1. **State the problem:** We are given three triangles ABE, ADE, and CBE on a coordinate grid with vertices at integer coordinates. We need to determine which statement about their congruence is true. 2. **List the coordinates:** - A(-4,-1) - B(-1,3) - D(1,-1) - E(0,1) - C(4,3) 3. **Calculate side lengths using the distance formula:** The distance between points $(x_1,y_1)$ and $(x_2,y_2)$ is $$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - For triangle ABE: - $AB = \sqrt{(-1 + 4)^2 + (3 + 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ - $BE = \sqrt{(0 + 1)^2 + (1 - 3)^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}$ - $AE = \sqrt{(0 + 4)^2 + (1 + 1)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$ - For triangle ADE: - $AD = \sqrt{(1 + 4)^2 + (-1 + 1)^2} = \sqrt{5^2 + 0^2} = 5$ - $DE = \sqrt{(0 - 1)^2 + (1 + 1)^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$ - $AE$ (already calculated) $= 2\sqrt{5}$ - For triangle CBE: - $CB = \sqrt{(-1 - 4)^2 + (3 - 3)^2} = \sqrt{(-5)^2 + 0^2} = 5$ - $BE$ (already calculated) $= \sqrt{5}$ - $CE = \sqrt{(0 - 4)^2 + (1 - 3)^2} = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$ 4. **Compare side lengths:** - Triangle ABE sides: $5$, $\sqrt{5}$, $2\sqrt{5}$ - Triangle ADE sides: $5$, $\sqrt{5}$, $2\sqrt{5}$ - Triangle CBE sides: $5$, $\sqrt{5}$, $2\sqrt{5}$ All three triangles have the same side lengths. 5. **Conclusion:** Since all three triangles have the same three side lengths, they are congruent by SSS (Side-Side-Side) congruence. **Final answer:** **D Triangle ABE, △ADE, and △CBE are all congruent.**