1. **State the problem:** Find the distance between the two points $(-4,7)$ and $(7,-5)$ on the coordinate plane. 2. **Formula used:** The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This formula comes from the Pythagorean theorem, where the difference in $x$ and $y$ coordinates form the legs of a right triangle. 3. **Apply the formula:** Calculate the differences: $$x_2 - x_1 = 7 - (-4) = 7 + 4 = 11$$ $$y_2 - y_1 = -5 - 7 = -12$$ 4. **Square the differences:** $$11^2 = 121$$ $$(-12)^2 = 144$$ 5. **Sum the squares:** $$121 + 144 = 265$$ 6. **Take the square root:** $$d = \sqrt{265}$$ 7. **Simplify the radical if possible:** Factor 265: $$265 = 5 \times 53$$ Neither 5 nor 53 is a perfect square, so $\sqrt{265}$ is already in simplest radical form. **Final answer:** $$\boxed{\sqrt{265}}$$