Step-by-step explanation:
The distance formula between two points:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Substitute the coordinates of the points.
[tex]A(4,\ 0),\ B(2,\ 1),\ C(-1,\ -5)\\\\AB=\sqrt{(2-4)^2+(1-0)^2}=\sqrt{(-2)^2+1^2}=\sqrt{4+1}=\sqrt5\\\\AC=\sqrt{(-1-4)^2+(-5-0)^2}=\sqrt{(-5)^2+(-5)^2}=\sqrt{25+25}=\sqrt{50}\\\\BC=\sqrt{(-1-2)^2+(-5-1)^2}=\sqrt{(-3)^2+(-6)^2}=\sqrt{9+36}=\sqrt{45}[/tex]
If a ≤ b < c are the sides of the right triangle, then
a² + b² = c²
[tex]\sqrt5<\sqrt{45}<\sqrt{50}\\\\(\sqrt5)^2+(\sqrt{45})^2=5+45=50\\\\(\sqrt{50})^2=50\\\\\bold{CORRECT}[/tex]
used [tex](\sqrt{a})^2=a[/tex] for a ≥ 0.
[tex]AB^2+BC^2=AC^2[/tex] therefore ΔABC is a right triangle.
