1. **Problem Statement:** Determine if the triangle with vertices at points $(2,3)$, $(4,9)$, and $(-2,7)$ is isosceles. 2. **Formula Used:** To check if a triangle is isosceles, we calculate the lengths of all three sides using the distance formula between two points $A(x_1,y_1)$ and $B(x_2,y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate the distances:** - Distance between $(2,3)$ and $(4,9)$: $$d_1 = \sqrt{(4-2)^2 + (9-3)^2} = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}$$ - Distance between $(4,9)$ and $(-2,7)$: $$d_2 = \sqrt{(-2-4)^2 + (7-9)^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}$$ - Distance between $(-2,7)$ and $(2,3)$: $$d_3 = \sqrt{(2+2)^2 + (3-7)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$$ 4. **Check for isosceles:** Since $d_1 = d_2 = 2\sqrt{10}$ and $d_3 = 4\sqrt{2}$, two sides are equal. 5. **Conclusion:** The triangle is isosceles because it has at least two sides of equal length. **Final answer:** The triangle with vertices $(2,3)$, $(4,9)$, and $(-2,7)$ is isosceles.