Key Concepts and Formulas

• Circumcenter of a Right-Angled Triangle: The circumcenter of a right-angled triangle is the midpoint of its hypotenuse.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Inradius of a Right-Angled Triangle: If the sides of the right angle are $a, b$ and the hypotenuse is $c$, the inradius is given by $r = \frac{a+b-c}{2}$.

Step 1: Identify the Type of Triangle

Let the vertices of $\triangle \mathrm{ABC}$ be $\mathrm{A}(\alpha, -2)$, $\mathrm{B}(\alpha, 6)$, and $\mathrm{C}\left(\frac{\alpha}{4}, -2\right)$.

Since the x-coordinates of A and B are the same ($\alpha$), side AB is parallel to the y-axis. Since the y-coordinates of A and C are the same (-2), side AC is parallel to the x-axis. Therefore, AB and AC are perpendicular, meaning $\angle \mathrm{BAC} = 90^\circ$. Thus, $\triangle \mathrm{ABC}$ is a right-angled triangle with the right angle at vertex A.

Why this step is taken: Identifying the triangle type simplifies later calculations. Knowing it's a right triangle allows us to use specific formulas for area, circumcenter, and inradius.

Step 2: Determine the Value of $\alpha$ using the Circumcenter Property

The circumcenter of $\triangle \mathrm{ABC}$ is given as $\left(5, \frac{\alpha}{4}\right)$. For a right-angled triangle, the circumcenter is the midpoint of the hypotenuse. Since angle A is the right angle, the hypotenuse is BC.

The midpoint of BC is: $\left(\frac{\alpha + \frac{\alpha}{4}}{2}, \frac{6 + (-2)}{2}\right) = \left(\frac{\frac{5\alpha}{4}}{2}, \frac{4}{2}\right) = \left(\frac{5\alpha}{8}, 2\right)$

We equate this to the given circumcenter: $\left(\frac{5\alpha}{8}, 2\right) = \left(5, \frac{\alpha}{4}\right)$

Equating the x-coordinates: $\frac{5\alpha}{8} = 5 \implies 5\alpha = 40 \implies \alpha = 8$

Equating the y-coordinates: $2 = \frac{\alpha}{4} \implies \alpha = 8$

Both equations give $\alpha = 8$.

Why this step is taken: Finding $\alpha$ is crucial as it determines the exact coordinates of the vertices, which are needed to calculate side lengths and other geometric properties.

Step 3: Calculate Vertices and Side Lengths

With $\alpha = 8$, the vertices are:

• $\mathrm{A}(8, -2)$
• $\mathrm{B}(8, 6)$
• $\mathrm{C}\left(\frac{8}{4}, -2\right) = \mathrm{C}(2, -2)$

Calculate the side lengths:

• $\mathrm{AB} = |6 - (-2)| = 8$
• $\mathrm{AC} = |8 - 2| = 6$
• $\mathrm{BC} = \sqrt{(8-2)^2 + (6-(-2))^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Why this step is taken: The side lengths are necessary for evaluating the area, perimeter, circumradius, and inradius.

Step 4: Evaluate Each Option

(A) Area of $\triangle$ABC is 24: Area $= \frac{1}{2} \times \mathrm{AB} \times \mathrm{AC} = \frac{1}{2} \times 8 \times 6 = 24$. This is correct.

(B) Perimeter of $\triangle$ABC is 25: Perimeter $= \mathrm{AB} + \mathrm{AC} + \mathrm{BC} = 8 + 6 + 10 = 24$. This is NOT correct.

(C) Circumradius of $\triangle$ABC is 5: Circumradius $= \frac{\mathrm{BC}}{2} = \frac{10}{2} = 5$. This is correct.

(D) Inradius of $\triangle$ABC is 2: Inradius $= \frac{\mathrm{AB} + \mathrm{AC} - \mathrm{BC}}{2} = \frac{8 + 6 - 10}{2} = \frac{4}{2} = 2$. This is correct.

Why this step is taken: This step verifies each option against the calculated values to determine which one is incorrect.

Step 5: Conclude the NOT Correct Statement

The only statement that is not correct is (B), which states the perimeter is 25, while the actual perimeter is 24.

Common Mistakes to Avoid:

• Failing to recognize the right-angled triangle, leading to more complex calculations.
• Arithmetic errors while calculating side lengths or area, perimeter, etc.
• Misreading the question and selecting a correct statement instead of the incorrect one.

Summary: The problem involves identifying a right-angled triangle, finding the value of an unknown coordinate using the circumcenter property, calculating side lengths, and then evaluating statements about the triangle's properties. The statement that is NOT correct is that the perimeter is 25.

The final answer is \boxed{B}.