Statement 1 : Let the vertices of a ` A B C` be `A(-5,-2),B(7,6),` and `C(5,-4)` . Then the coordinates of the circumcenter are `(1,2)dot` Statement 2 : In a right-angled triangle, the midpoint of the hypotenuse is the circumcenter of the triangle.

Answer

To solve the problem, we need to verify the two statements regarding the triangle formed by the vertices A(-5, -2), B(7, 6), and C(5, -4). We will check if the triangle is a right triangle and then find the circumcenter.

### Step 1: Find the slopes of the sides of the triangle

1. **Calculate the slope of AB:**
\[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-2)}{7 - (-5)} = \frac{6 + 2}{7 + 5} = \frac{8}{12} = \frac{2}{3} \]

2. **Calculate the slope of BC:**
\[ \text{slope of } BC = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-4)}{7 - 5} = \frac{6 + 4}{7 - 5} = \frac{10}{2} = 5 \]

3. **Calculate the slope of AC:**
\[ \text{slope of } AC = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-4)}{-5 - 5} = \frac{-2 + 4}{-10} = \frac{2}{-10} = -\frac{1}{5} \]

### Step 2: Check if the triangle is a right triangle

To check if the triangle is a right triangle, we need to see if the product of the slopes of two sides is -1 (indicating they are perpendicular).

- Check slopes of BC and AC:
\[ \text{slope of } BC \times \text{slope of } AC = 5 \times -\frac{1}{5} = -1 \]
This confirms that BC is perpendicular to AC, hence triangle ABC is a right triangle.

### Step 3: Find the circumcenter

In a right triangle, the circumcenter is located at the midpoint of the hypotenuse. The hypotenuse in triangle ABC is AB.

1. **Find the midpoint of AB:**
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-5 + 7}{2}, \frac{-2 + 6}{2} \right) = \left( \frac{2}{2}, \frac{4}{2} \right) = (1, 2) \]

### Conclusion

- The circumcenter of triangle ABC is at (1, 2).
- Since triangle ABC is a right triangle, the midpoint of the hypotenuse (AB) is indeed the circumcenter.

### Final Statements

- **Statement 1**: True, the coordinates of the circumcenter are (1, 2).
- **Statement 2**: True, in a right-angled triangle, the midpoint of the hypotenuse is the circumcenter.