Let $$A(\alpha, -2)$$, $$B(\alpha, 6)$$ and $$C\left(\frac{\alpha}{4}, -2\right)$$ be vertices of a $$\Delta ABC$$. If $$\left(5, \frac{\alpha}{4}\right)$$ is the circumcentre of $$\Delta ABC$$, then which of the following is NOT correct about $$\Delta ABC$$

We have $$A(\alpha, -2)$$, $$B(\alpha, 6)$$, and $$C\left(\dfrac{\alpha}{4}, -2\right)$$ as vertices of $$\triangle ABC$$, with circumcentre at $$\left(5, \dfrac{\alpha}{4}\right)$$.

We observe that $$A$$ and $$B$$ have the same $$x$$-coordinate $$\alpha$$, so $$AB$$ is vertical with length $$|6 - (-2)| = 8$$. Also, $$A$$ and $$C$$ have the same $$y$$-coordinate $$-2$$, so $$AC$$ is horizontal with length $$\left|\alpha - \dfrac{\alpha}{4}\right| = \dfrac{3|\alpha|}{4}$$. Since $$AB \perp AC$$, this is a right triangle with the right angle at $$A$$.

For a right triangle, the circumcentre is the midpoint of the hypotenuse $$BC$$. The midpoint of $$BC$$ is $$\left(\dfrac{\alpha + \frac{\alpha}{4}}{2},\; \dfrac{6 + (-2)}{2}\right) = \left(\dfrac{5\alpha}{8},\; 2\right)$$.

Setting this equal to the given circumcentre $$\left(5, \dfrac{\alpha}{4}\right)$$: from the $$y$$-coordinate, $$\dfrac{\alpha}{4} = 2$$, so $$\alpha = 8$$. Let us verify with the $$x$$-coordinate: $$\dfrac{5 \cdot 8}{8} = 5$$. This checks out.

Now with $$\alpha = 8$$: $$A = (8, -2)$$, $$B = (8, 6)$$, $$C = (2, -2)$$, and the circumcentre is $$(5, 2)$$.

We compute the sides: $$AB = 8$$ (vertical), $$AC = 8 - 2 = 6$$ (horizontal), and $$BC = \sqrt{(8-2)^2 + (6-(-2))^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$.

Now we check each option:

Area: $$\dfrac{1}{2} \times AB \times AC = \dfrac{1}{2} \times 8 \times 6 = 24$$. This is correct.

Perimeter: $$8 + 6 + 10 = 24$$, not 25. So "perimeter is 25" is NOT correct.

Circumradius: For a right triangle, $$R = \dfrac{\text{hypotenuse}}{2} = \dfrac{10}{2} = 5$$. This is correct.

Inradius: $$r = \dfrac{\text{Area}}{s}$$ where $$s = \dfrac{24}{2} = 12$$. So $$r = \dfrac{24}{12} = 2$$. This is correct.

Hence, the correct answer is Option B: perimeter is 25 (this statement is NOT correct).